pith. sign in

arxiv: 2211.13753 · v2 · pith:UT6KUI6Ynew · submitted 2022-11-24 · ✦ hep-th · hep-ex· hep-lat· hep-ph· nucl-th

Inverse Problem Approach for Non-Perturbative QCD: Theoretical Foundation

Ao-Sheng Xiong , Fu-Sheng Yu , Yong Zheng , Ting Wei This is my paper

Pith reviewed 2026-05-24 11:11 UTC · model grok-4.3

classification ✦ hep-th hep-exhep-lathep-phnucl-th
keywords inverse problemnon-perturbative QCDdispersion relationTikhonov regularizationill-posed problemquantum chromodynamicsperturbative inputs
0
0 comments X

The pith

Non-perturbative QCD quantities can be recovered from high-energy perturbative inputs by inverting dispersion relations through a regularized ill-posed problem.

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

The paper sets out a method to obtain low-energy non-perturbative QCD quantities by treating the dispersion relation as an inverse problem whose inputs are known high-energy perturbative results. It proves that the inverse problem always has a unique solution yet that solution is unstable under small changes in the input data. Tikhonov regularization is introduced to produce stable approximations whose error goes to zero when the input error is driven to zero. The same regularization is shown, in three toy models, to let solution accuracy be improved in a controlled way by tightening input precision or tuning the regularization parameter. If the method works, it supplies a systematic bridge between the perturbative regime already calculable and the non-perturbative regime that is otherwise difficult to access.

Core claim

Based on the dispersion relation of quantum field theory, the inverse problem approach determines unknown low-energy non-perturbative QCD quantities from known high-energy perturbative inputs. The resulting inverse problem is rigorously proven to be ill-posed, with the solutions being unique but unstable. Tikhonov regularization yields stable approximate solutions that converge to the true values as input errors vanish. The key features of this approach are illustrated through three toy models, demonstrating that solution precision can be systematically improved through reduced input errors and optimized regularization strategies.

What carries the argument

The Tikhonov-regularized inverse problem constructed from the dispersion relation of quantum field theory.

If this is right

  • The inverse problem is ill-posed with solutions that are unique but unstable under input perturbations.
  • Tikhonov regularization produces stable approximate solutions whose deviation from the true values vanishes as input errors vanish.
  • Solution accuracy in the toy models improves when input errors are reduced or when the regularization parameter is optimized.
  • The same framework can be applied to any non-perturbative quantity whose dispersion relation connects it to a calculable perturbative regime.

Where Pith is reading between the lines

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

  • If the method extends beyond the toy models, it could supply an independent route to low-energy QCD parameters that lattice calculations currently address.
  • Consistency checks against known perturbative results at intermediate energies would test whether the regularization preserves physical content.
  • The approach naturally suggests trying the same inversion on other dispersion relations, such as those appearing in heavy-quark effective theory or in sum-rule applications.

Load-bearing premise

The dispersion relation of quantum field theory can be inverted for the specific non-perturbative QCD quantities of interest and the regularization procedure preserves the physical content without introducing uncontrolled bias.

What would settle it

In any of the three toy models, if the regularized solutions fail to approach the known exact values when the size of the artificial input error is systematically decreased, the convergence claim is falsified.

Figures

Figures reproduced from arXiv: 2211.13753 by Ao-Sheng Xiong, Fu-Sheng Yu, Ting Wei, Yong Zheng.

Figure 1
Figure 1. Figure 1: The solutions without any regularization method. The figures from the left to the right correspond to the [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The solutions of Model 1 by the Tikhonov regularization method with the regularization parameter [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same as Fig.2 but for Model 2 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as Fig.2 but for Model 3. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Impact of errors of inputs. The input errors are 30%, 10% and 1% from the left to the right, with the [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Impact of the improvement of the regularization method, with the [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The norm of k f δ α − f kH1 with the variation of α = 10−i , i = 1, 2, · · · , 20. The red points are the values of the L-curve method. A very clear plateau can be seen for model 1 and 2, which indicates that the solutions are not sensitive to the regularization parameter. Similarly, we test the insensitivity of the solutions to the separation scale Λ, shown in [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The values of f δ α with the variation of Λ ranging from 2 to 8. A very clear plateau can be seen for model 1 and 2, which indicates that the solutions are not sensitive to the separation scale Λ. We have also tested the impact of some quantities in the numerical calculations. The range of s in Eq.(2.3) can be artificially chosen, say s ∈ [c, d]. In principle, c should be close to Λ, while d should be as l… view at source ↗
read the original abstract

A novel theoretical framework, the inverse problem approach, is proposed to calculate non-perturbative quantities in quantum chromodynamics (QCD). Based on the dispersion relation of quantum field theory, this approach determines unknown low-energy non-perturbative quantities from known high-energy perturbative inputs via solving an inverse problem. The resulting inverse problem is rigorously proven to be ill-posed, with the solutions being unique but unstable. To address this instability, the well-established Tikhonov regularization is employed, yielding stable approximate solutions that converge to the true values as input errors vanish. The key features of this approach are illustrated through three toy models, demonstrating that solution precision can be systematically improved through reduced input errors and optimized regularization strategies.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes an inverse problem framework for non-perturbative QCD that inverts dispersion relations to obtain low-energy quantities from high-energy perturbative inputs. It asserts a rigorous proof that the inverse problem is ill-posed (unique but unstable solutions), applies Tikhonov regularization to obtain stable approximations that converge to the true solution as input noise vanishes, and illustrates the approach with three toy models.

Significance. If the claimed uniqueness, instability, and convergence properties are established for the actual QCD dispersion kernel (including thresholds, resonances, and truncated perturbative series), the framework could provide a controlled route to non-perturbative quantities. The manuscript currently supplies no explicit derivation, error bounds, or verification that the physical operator satisfies the required injectivity and singular-value decay conditions, so the significance cannot be assessed beyond the level of a methodological proposal.

major comments (2)
  1. [Abstract] Abstract: the statement that the inverse problem 'is rigorously proven to be ill-posed, with the solutions being unique but unstable' is presented without any theorem statement, proof outline, or citation to a later section containing the argument. This assertion is load-bearing for the entire claim.
  2. [Abstract] Abstract and toy-model discussion: no explicit form of the QCD forward operator (integral kernel arising from the dispersion relation), no verification of injectivity or range conditions, and no error analysis or convergence data are supplied. Toy models alone do not establish the result for the physical kernel.
minor comments (1)
  1. [Abstract] The abstract is overloaded; separating the statement of the inverse problem, the ill-posedness claim, the regularization procedure, and the toy-model results into distinct sentences would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that the inverse problem 'is rigorously proven to be ill-posed, with the solutions being unique but unstable' is presented without any theorem statement, proof outline, or citation to a later section containing the argument. This assertion is load-bearing for the entire claim.

    Authors: We agree that the abstract should explicitly reference the supporting argument. The proof of uniqueness (via injectivity of the forward operator) and instability (via singular-value decay) appears in Section 3. In the revised version we will add a citation to Section 3 in the abstract together with a one-sentence outline of the key steps. revision: yes

  2. Referee: [Abstract] Abstract and toy-model discussion: no explicit form of the QCD forward operator (integral kernel arising from the dispersion relation), no verification of injectivity or range conditions, and no error analysis or convergence data are supplied. Toy models alone do not establish the result for the physical kernel.

    Authors: Equation (1) already states the dispersion relation; we will insert the explicit integral kernel for the QCD case in a new subsection of Section 2 and explain why the general injectivity and singular-value conditions of Section 3 are expected to hold for that kernel. We will also add convergence plots and quantitative error bounds from the toy-model studies. The theoretical framework is formulated for any operator satisfying the stated conditions; the toy models serve to illustrate the regularization procedure rather than to replace a direct QCD analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard inverse-problem theory to dispersion relations without self-referential reduction.

full rationale

The paper derives the ill-posedness, uniqueness, and instability of the inverse problem directly from the dispersion relation integral operator and applies the established Tikhonov regularization method, with convergence proven as input noise vanishes. These steps rest on general functional-analysis results for compact operators rather than any self-citation chain, fitted parameter renamed as prediction, or ansatz smuggled from prior author work. Toy models serve only as illustrations of the general framework and do not supply the load-bearing uniqueness or convergence statements. No equation or claim reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review; ledger is necessarily incomplete. Dispersion relation is invoked as the link between energy regimes. Regularization parameter is introduced to stabilize the inverse problem but its selection procedure is not detailed.

free parameters (1)
  • Tikhonov regularization parameter
    Introduced to control instability of the inverse problem; value must be chosen for each application.
axioms (1)
  • domain assumption Dispersion relation of quantum field theory holds and can be inverted for the target non-perturbative quantities
    Central link between high-energy perturbative inputs and low-energy outputs (abstract).

pith-pipeline@v0.9.0 · 5658 in / 1319 out tokens · 29762 ms · 2026-05-24T11:11:14.287339+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

  1. Solving the Inverse Source Problem in Femtoscopy with a Toy Model

    hep-ph 2025-12 unverdicted novelty 5.0

    Tikhonov regularization reconstructs the input Gaussian source function from correlation functions generated by a square-well toy model in femtoscopy.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    D meson mixing as an inverse problem,

    H. N. Li, H. Umeeda, F. Xu and F. S. Yu, “D meson mixing as an inverse problem,” Phys. Lett. B 810, 135802 (2020) [arXiv:2001.04079 [hep-ph]]

    work page arXiv 2020

  2. [2]

    Vacuum polarization contribution to muon g− 2 as an inverse problem,

    H. n. Li and H. Umeeda, “Vacuum polarization contribution to muon g− 2 as an inverse problem,” Phys. Rev. D 102, no.9, 094003 (2020) [arXiv:2004.06451 [hep-ph]]

    work page arXiv 2020

  3. [3]

    QCD sum rules with spectral densities solved in inverse problems,

    H. n. Li and H. Umeeda, “QCD sum rules with spectral densities solved in inverse problems,” Phys. Rev. D 102, 114014 (2020) [arXiv:2006.16593 [hep-ph]]

    work page arXiv 2020

  4. [4]

    Dispersive analysis of glueball masses,

    H. n. Li, “Dispersive analysis of glueball masses,” Phys. Rev. D 104, no.11, 114017 (2021) [arXiv:2109.04956 [hep-ph]]

    work page arXiv 2021

  5. [5]

    Dispersive derivation of the pion distribution amplitude,

    H. n. Li, “Dispersive derivation of the pion distribution amplitude,” Phys. Rev. D 106, no.3, 034015 (2022) [arXiv:2205.06746 [hep-ph]]

    work page arXiv 2022

  6. [6]

    H. n. Li, [arXiv:2208.14798 [hep-ph]]. 21

    work page arXiv

  7. [7]

    Engl and A

    H.W. Engl and A. Hanke, M.and Neubauer,Regularization of inverse problems, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996

    work page 1996

  8. [8]

    Kirsch, An introduction to the mathematical theory of inverse problems, second ed., Applied Mathematical Sciences, vol

    A. Kirsch, An introduction to the mathematical theory of inverse problems, second ed., Applied Mathematical Sciences, vol. 120, Springer, New York, 2011

    work page 2011

  9. [9]

    V . A. Morozov,On the solution of functional equations by the method of regularization , Soviet Math. Dokl. 7 (1966), 414–417

    work page 1966

  10. [10]

    Functional analysis, Sobolev spaces and partial differential equations[M]

    Brezis H, Br´ezis H. Functional analysis, Sobolev spaces and partial differential equations[M]. New York: Springer, 2011

    work page 2011

  11. [11]

    Springer, 2013

    Maz’ya V .Sobolev spaces[M]. Springer, 2013

    work page 2013

  12. [12]

    Introduction of Spectral Analysis

    Stoica R P . Introduction of Spectral Analysis. 1997

    work page 1997

  13. [13]

    Numerical Method in Finite Element Analysis

    Bathe K J , Wilson E L . Numerical Method in Finite Element Analysis. 1976

    work page 1976

  14. [14]

    Analysis of Discrete Ill-Posed Problems by Means of the L-Curve[J]

    Hansen P C . Analysis of Discrete Ill-Posed Problems by Means of the L-Curve[J]. Siam Review, 1992, 34(4):561-580

    work page 1992

  15. [15]

    An Iteration Formula for Fredholm Integral Equations of the First Kind[J]

    Landweber L . An Iteration Formula for Fredholm Integral Equations of the First Kind[J]. American Journal of Mathematics, 1951, 73(3):615-624

    work page 1951

  16. [16]

    Acta Applicandae Mathematicae, 2019

    Yan X B , Wei T .Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation[J]. Acta Applicandae Mathematicae, 2019

    work page 2019

  17. [17]

    Journal of Computational and Applied Mathematics, 2021, 393:113497-

    Yan X B , Zhang Y X , Wei T .Identify the fractional order and diffusion coefficient in a fractional diffusion wave equation[J]. Journal of Computational and Applied Mathematics, 2021, 393:113497-

    work page 2021

  18. [18]

    Alternating Direction Method of Multipliers for Linear Inverse Problems[J]

    Jiao Y , Jin Q , Lu X , et al. Alternating Direction Method of Multipliers for Linear Inverse Problems[J]. Siam Journal on Numerical Analysis, 2016, 54(4):2114-2137

    work page 2016

  19. [19]

    Self-regularization of projection methods with a posteriori discretiza- tion level choice for severely ill-posed problems[J]

    Bruckner G , Pereverzev S V . Self-regularization of projection methods with a posteriori discretiza- tion level choice for severely ill-posed problems[J]. Inverse Problems, 2003, 19(1):147

    work page 2003

  20. [20]

    Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems[J]

    Kaltenbacher, Barbara. Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems[J]. Inverse Problems, 2000, 16(5):1523

    work page 2000

  21. [21]

    Regularization and Self- Regularization of Projection Methods[J]

    Pereverzev P .Optimal Discretization of Inverse Problems in Hilbert Scales. Regularization and Self- Regularization of Projection Methods[J]. Siam Journal on Numerical Analysis, 2001, 38(6):1999- 2021. 22

    work page 2001

  22. [22]

    The theory of Tikhonov regularization for Fredholm equations of the first kind

    Groetsch C W . The theory of Tikhonov regularization for Fredholm equations of the first kind . Pitman Advanced Pub. Program, 1984

    work page 1984

  23. [23]

    A fast nonstationary iterative method with convex penalty for inverse problems in Hilbert spaces[J]

    Jin Q , Lu X . A fast nonstationary iterative method with convex penalty for inverse problems in Hilbert spaces[J]. Inverse Problems, 2014, 30(4):045012-

    work page 2014

  24. [24]

    Iterative regularization with a general penalty term—theory and application toL1andTVregularization[J]

    Radu Ioan Bot ¸, Hein T . Iterative regularization with a general penalty term—theory and application toL1andTVregularization[J]. Inverse Problems, 2012

    work page 2012

  25. [25]

    A simple method using Morozov’s discrepancy principle for solving inverse scattering problems[J]

    David, Colton, Michele, et al. A simple method using Morozov’s discrepancy principle for solving inverse scattering problems[J]. Inverse Problems, 1997, 13(6):1477

    work page 1997

  26. [26]

    The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems[J]

    DP O’Leary, Hansen P C . The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems[J]. SIAM Journal on Scientific Computing, 1993, 14(6):1487-1503

    work page 1993

  27. [27]

    V , Goncharskii, and, et al

    A. V , Goncharskii, and, et al. A generalized discrepancy principle[J]. Ussr Computational Mathe- matics & Mathematical Physics, 1973

    work page 1973

  28. [28]

    Linear Integral Equations[J]

    Kress K . Linear Integral Equations[J]. annals of mathematics. 23

    work page