arxiv: 2512.06904 · v2 · submitted 2025-12-07 · ✦ hep-ph

Solving the Inverse Source Problem in Femtoscopy with a Toy Model

Ao-Sheng Xiong , Qi-Wei Yuan , Ming-Zhu Liu , Fu-Sheng Yu , Zhi-Wei Liu , Li-Sheng Geng This is my paper

Pith reviewed 2026-05-17 00:27 UTC · model grok-4.3

classification ✦ hep-ph

keywords femtoscopyinverse source problemTikhonov regularizationhadron correlation functionssource function reconstructionsquare-well potentialtoy modelheavy-ion collisions

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The pith

Tikhonov regularization reconstructs Gaussian source functions from hadron correlation functions in a square-well toy model.

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

Femtoscopy extracts hadron interactions from momentum correlation functions, but these functions mix unknown source shapes with known wave functions, leaving the source ambiguous and usually fixed to a Gaussian. The paper builds a toy model that generates correlation functions from a square-well potential of varying strengths and then inverts them with Tikhonov regularization to recover the original source. When the input is Gaussian the method recovers it accurately; the same pipeline is also tested on a hybrid source. Success in the controlled toy setting suggests the regularization step can be applied to real data to extract non-Gaussian sources without assuming their form in advance.

Core claim

A toy model based on the Tikhonov regularization is used to solve the inverse source problem: correlation functions are first computed from known Gaussian and hybrid sources folded with square-well wave functions, then the regularization inverts those functions to reconstruct the input sources. The Gaussian source is recovered successfully for four different well depths, demonstrating that the approach can in principle extract realistic source functions of hadron pairs from experimental correlation data.

What carries the argument

Tikhonov regularization applied to deconvolve the source function from the measured correlation function after the square-well wave function is known.

If this is right

Gaussian sources are recovered accurately across four different square-well strengths.
The same pipeline can be applied to hybrid source forms to test robustness.
Once validated, the method supplies data-driven source functions that reduce reliance on the Gaussian approximation in femtoscopy analyses.
Improved sources allow tighter constraints on the underlying hadron-hadron interaction potentials.

Where Pith is reading between the lines

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

If the method works on real data it would let experiments report source shapes directly rather than assume them, tightening interaction extractions.
The regularization could be combined with detector-response modeling to handle realistic experimental noise.
Extending the toy model to more realistic potentials such as Coulomb plus strong interaction would be a direct next test.

Load-bearing premise

The regularization parameter tuned on the clean square-well toy model remains stable and unbiased when real experimental correlation functions include detector effects, non-Gaussian sources, and more complicated potentials.

What would settle it

Reconstruct the source from actual heavy-ion collision correlation data for a well-studied pair such as proton-proton and compare the result against an independent, non-Gaussian source shape obtained from transport simulations or other observables.

Figures

Figures reproduced from arXiv: 2512.06904 by Ao-Sheng Xiong, Fu-Sheng Yu, Li-Sheng Geng, Ming-Zhu Liu, Qi-Wei Yuan, Zhi-Wei Liu.

**Figure 1.** Figure 1: FIG. 1: Unregularized solutions (red curves) for the four sources, exhibiting unstable reconstructions that deviate [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2: Reconstructed solutions [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Reconstructed solutions [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Reconstruction of the source [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

Hadron-hadron interactions, as a non-perturbative effect, play a significant role in understanding phenomenological problems in particle physics. Femtoscopy is a powerful tool in heavy-ion collision experiments, enabling the extraction of hadron-hadron interactions via momentum-correlation functions (CFs). These CFs are generally factorized into a convolution of source functions and hadron-hadron wave functions, with the latter encoding information about hadron-hadron interactions. However, source functions remain ambiguous and are commonly approximated by a Gaussian form. Reconstructing source functions from experimental correlation data constitutes an ``inverse problem." To address it, we propose a toy model based on the Tikhonov regularization. Employing a square potential well of four distinct potential strengths, we calculate the CFs for inputs of a Gaussian source function and its hybrid form. The obtained CFs are subsequently used to reconstruct the source functions via the Tikhonov regularization. Our results demonstrate that the Gaussian source function can be successfully reconstructed, indicating the potential of this approach for extracting realistic source functions of hadron pairs of interest in the future.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Tikhonov regularization reconstructs the Gaussian source in this femtoscopy toy model, but the parameter choice depends on knowing the true source in advance.

read the letter

The main thing to know is that Tikhonov regularization successfully reconstructs the input Gaussian source from correlation functions in this square-well toy model for femtoscopy, but the regularization parameter is tuned with knowledge of the true source, which raises questions about its use on real data. The paper sets up the inverse problem clearly. They use a square potential well with varying strengths to compute correlation functions for a Gaussian source and a hybrid version. Then they apply Tikhonov regularization to recover the source function from those CFs. It works for the Gaussian case, as stated in the abstract. This is a straightforward numerical test that shows regularization can handle the ill-posed nature of source reconstruction without assuming a form upfront. What stands out is the clean validation: since the inputs are known, you can directly compare the output to the original source. That's better than many inverse problem papers that lack such checks. The choice of Tikhonov is appropriate here because it adds a smoothness penalty that stabilizes the solution. On the downside, the stress test note is on point. The paper does not appear to provide an automatic way to pick the regularization parameter for cases where the source shape is not known in advance. In the toy runs, lambda is adjusted until the Gaussian is recovered, but real experimental correlation functions come with noise, detector effects, and potentially complicated sources. A data-driven selection method would make the claim stronger. Also, the results for the hybrid source are mentioned but without details on how well it performs or any error quantification. Overall, this is for specialists in heavy-ion femtoscopy who want to explore alternatives to the Gaussian source assumption. It could spark ideas for better source extraction in future analyses. The work shows clear thinking on the problem and engages with the standard tools for inverse problems. It deserves peer review to get feedback on extending the method to more realistic scenarios.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a Tikhonov-regularization approach to solve the inverse source problem in femtoscopy. Using a square-well toy potential of four strengths, the authors generate correlation functions from known Gaussian and hybrid source inputs and then reconstruct the sources, claiming that the Gaussian source is successfully recovered.

Significance. A robust, data-driven method for source reconstruction would reduce reliance on the Gaussian ansatz and improve extraction of hadron-hadron interactions from measured correlation functions. The present toy-model demonstration is a necessary first step, but its evidential value is limited by the absence of quantitative accuracy metrics and by the use of oracle-tuned regularization parameters.

major comments (2)

[Numerical results / reconstruction section] The Tikhonov regularization parameter λ is chosen so that the reconstructed source matches the known input Gaussian (see the numerical reconstruction procedure). This oracle selection does not supply a procedure (L-curve, discrepancy principle, or cross-validation) that could be applied when the true source is unknown, as required for experimental data. The central claim of applicability therefore rests on an untested assumption about parameter stability.
[Abstract and Results] No quantitative reconstruction metrics (L2-norm error, point-wise residuals, or χ² between input and reconstructed sources) are reported, nor is any comparison presented with other regularizers or with the hybrid-source case. The abstract statement that the Gaussian source “can be successfully reconstructed” therefore lacks the numerical support needed to assess the method’s performance.

minor comments (2)

Define the precise functional form of the “hybrid” source used in the calculations.
[Introduction] Add references to existing literature on inverse problems and regularization techniques applied to femtoscopic correlation functions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the scope and limitations of our toy-model demonstration. We address each major point below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

Referee: [Numerical results / reconstruction section] The Tikhonov regularization parameter λ is chosen so that the reconstructed source matches the known input Gaussian (see the numerical reconstruction procedure). This oracle selection does not supply a procedure (L-curve, discrepancy principle, or cross-validation) that could be applied when the true source is unknown, as required for experimental data. The central claim of applicability therefore rests on an untested assumption about parameter stability.

Authors: We agree that oracle tuning of λ is used in the present toy-model study because the true source is known by construction. This choice was made solely to verify that the regularization can recover the input under controlled conditions. For experimental application the parameter must be selected without knowledge of the source. In the revised manuscript we will add a dedicated subsection describing and applying two data-driven selection methods (L-curve and generalized cross-validation) to the same toy-model correlation functions, thereby demonstrating a practical route that does not rely on oracle information. revision: yes
Referee: [Abstract and Results] No quantitative reconstruction metrics (L2-norm error, point-wise residuals, or χ² between input and reconstructed sources) are reported, nor is any comparison presented with other regularizers or with the hybrid-source case. The abstract statement that the Gaussian source “can be successfully reconstructed” therefore lacks the numerical support needed to assess the method’s performance.

Authors: We accept that quantitative error measures would strengthen the assessment of reconstruction quality. Although the current figures illustrate visual agreement, the revised manuscript will report L2-norm errors, point-wise residuals, and χ² values for both the Gaussian and hybrid source reconstructions. A short comparison of reconstruction fidelity between the two source types will also be included. These additions will supply the numerical evidence supporting the abstract claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper generates correlation functions from known Gaussian (and hybrid) source inputs convolved with square-well potentials, then applies standard Tikhonov regularization to reconstruct the source and verifies success against the external ground-truth input. This constitutes an independent validation benchmark rather than a self-referential definition or fit. No load-bearing self-citations, self-definitional equations, or ansatz smuggling are present in the abstract or described method; the central claim remains a proof-of-concept demonstration of an established inverse-problem technique on controlled toy data.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard convolution relation between source and wave function plus the choice of a single regularization parameter whose value is not reported.

free parameters (1)

Tikhonov regularization parameter
Controls the trade-off between data fit and solution smoothness; its specific value is not given in the abstract.

axioms (1)

domain assumption The measured correlation function is the convolution of an unknown source function with the known interaction wave function.
Standard factorization used throughout femtoscopy literature.

pith-pipeline@v0.9.0 · 5507 in / 1230 out tokens · 55235 ms · 2026-05-17T00:27:42.122422+00:00 · methodology

discussion (0)

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We employ the Tikhonov regularization... minimize ||KS−Cδ||²l2 + α||LS||²l2 (Eq. 7); L-curve criterion for α
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Koonin-Pratt formula C(k)=∫d³r S12(r)|Ψ(r,k)|² (Eq. 1)

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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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.

Reconstructing rare particle source by femtoscopic correlations
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A statistical method reconstructs single-particle emission sources for rare particles directly from conditioned correlation kernels in femtoscopy, demonstrated on simulated J/ψ sources in pp collisions with 13% system...

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