Nucleon Wave Functions from QCD Vacuum Topology

Ismail Zahed; Wei-Yang Liu

REVIEW

Nucleon Wave Functions from QCD Vacuum Topology

Wei-Yang Liu, Ismail Zahed

High Energy Physics - Phenomenology · Submitted 29 Oct 2026

Pith's verdict: A genuinely new ILM diquark-kernel derivation with the right logical order, but the headline mass and g_A sit on a static/pole reduction that is never benchmarked in exactly the regime where it is least safe.

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arXiv ID2610.39215 (version 2)

Pith IDILLCFDH5, Pith's permanent record of this paper and its review

CitationIsmail Zahed, Wei-Yang Liu. "Nucleon Wave Functions from QCD Vacuum Topology." arXiv:2610.39215 (2026). https://pith.science/paper/ILLCFDH5

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Editor's note

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Two things to know. First, the derivation of the single-instanton and instanton-anti-instanton molecular diquark kernels for Nc=3, and their embedding in a permutation-symmetric quark-diquark Faddeev equation, is genuinely new work, and the logical order is right: the couplings GI and GIA are fixed by the pion and rho masses before the baryon sector, and the nucleon outputs are not used to fit anything. The circularity burden is low. Second, the headline numbers, m_N = 939.53 MeV and g_A = 1.265, rest on a static/pole reduction that is never benchmarked against the full momentum-dependent equation (29), in precisely the regime where the approximation is least safe. The axial-vector diquark pole sits 1.5 MeV below threshold, and this channel supplies essentially all of the 9.2 MeV mass shift from the scalar-only value and about two-thirds of g_A. A referee should not take those digits as predictions until that reduction is checked or error-bounded.

What the paper does well: the diquark spectrum computation is explicit and the comparisons with lattice and DSE numbers are honest, including the note that the ILM diquarks come out light because confinement is absent. The 98/2 scalar/axial split is disclosed as basis-dependent, and the distinction between pole weights and the light-front norm overlaps P(ss), P(sa), P(aa) is explained carefully: a 34% axial-containing light-front norm coexisting with a 2% pole weight is a genuine permutation-recoupling effect, not a sleight of hand. The near-cancellation alpha_2 ~ alpha_3/sqrt(3) matching Ref. [24] is a good consistency check. The authors also state plainly what the model cannot do: no Delta bound state, leading-Fock-only, and the helicity/sea discrepancies are assigned to higher Fock sectors.

Soft spots, in order of size. The static approximation S(k) -> -i/M and the pole replacement of the diquark propagators convert a barely-bound, near-continuum correlation into a zero-width pole; Appendix H itself calls the pole step a simple proof of principle. No comparison with the full equation is provided, so the error in m_N and g_A from this reduction is unknown, and it acts on the channel that controls both central results. Presenting m_N to 0.01 MeV is not supported when the approximation error is plausibly a few MeV; the 1.3 MeV proximity to the PDG value is quoted at a tolerance the calculation cannot sustain. The PDF evolution is fairly sensitive to alpha_s(mu0) over the 0.87-1.78 range the authors scan, which is honest but makes the lattice/global-fit comparisons looser than the tables suggest. Minor: no code or data accompany the paper, so the bubble integrals and the Faddeev diagonalization cannot be independently audited, and Table III lists a spectral-fit ILM scalar diquark mass of 420+-30 MeV from Ref. [13] with no comment on the 170 MeV gap from this paper's 591.63 MeV.

Bottom line: coherent, serious model calculation with a genuine new derivation. Send it to a serious referee with the instruction that the static/pole reduction needs a benchmark or an error estimate before the mass and g_A claims are taken at face value. Not a desk reject; a conditional accept with teeth.

Referee report

Significance

The logical structure of the paper is commendable: GI and GIA are fixed by the pion and rho masses (Table I), M and ρ are taken from earlier published ILM work, λ_N is fixed by the wave-function normalization (Eq. 43), and the nucleon mass, g_A, and PDFs are uncontaminated outputs. The derivations are explicit and checkable, the diquark correlation functions are given in closed form (App. G), the S3 permutation structure is handled systematically, and the axial charge passes an internal consistency check (∆u − ∆d = 1.265 at both the model scale and 2 GeV). If the approximation chain is controlled, this is a genuinely economical unification: the same vacuum input that fixes the pion and rho masses would predict a nucleon within ~1.3 MeV of experiment, the correct diquark hierarchy, g_A within ~1%, and valence PDFs comparable to lattice and global determinations, with the deviations in helicity and sea observables plausibly attributable to the missing higher Fock components that the authors identify. The principal concern is therefore not circularity, but the control and honest reporting of the systematic uncertainty of the BSF reduction.

Major objections (2)

[Sec. III and App. H; Eqs. (29)–(31), (22)/(H3); Tables III, V, VIII] The reduction of the three-body BSF equation is the load-bearing step for every headline number, and it is never benchmarked. In Eq. (31) the exchange kernel is made static (S(k) → −i/M) and the diquark propagators (20)–(21) are replaced by zero-width poles (22)/(H3), while the resulting vertex coefficients α_i (Eq. 32) feed the light-front wave functions (46)–(47), g_A, and the PDFs. The axial-vector channel, whose pole at 818.5 MeV sits only 1.5 MeV below the 2M = 820 MeV threshold, supplies the entire 9.2 MeV mass shift reported in Sec. III (sentence b129) and 0.822 of the total g_A = 1.265 (Table VIII); near threshold is exactly the regime where replacing a near-continuum correlation by a single pole is least justified. Appendix H itself labels the pole step 'a simple proof of principle', and neither a comparison with Eq. (29) using the untruncated propagators (20)–(21) nor a systematic variation of the static replacement is provided. Because the uncontrolled ingredient contributes 9.2 MeV and 0.82 in g_A while the claimed agreement with experiment is 1.3 MeV and ~0.01, the stated precision of m_N = 939.53 MeV and g_A = 1.265 is not supported by the manuscript as it stands. I ask the authors to solve Eq. (29) with momentum-dependent S(k) and the full diquark propagators in at least one benchmark configuration, or to estimate the error by a controlled variation (for instance, retaining the leading momentum dependence in the exchange kernel or including the finite width of the near-threshold axial diquark), and then to re-quote m_N, g_A, and the coefficients α_i with digits and uncertainties consistent with the achieved accuracy.
[Sec. IV; Eqs. (35) and (52)–(53); Tables VI and VIII] The relation between the pole weights (w_s = 98.0%, w_a = 2.0%, Eq. (35)) and the fully symmetrized light-front decomposition (P(ss) = 0.305, P(sa) = 0.350, P(aa) = 0.339, Table VI) needs sharper quantitative support. The axial-vector configuration carries about one third of the light-front norm and, through scalar–axial and permutation interference, about 65% of g_A (Table VIII), so the abstract's '98%/2%' phrasing is easy to misread as a statement about the composition of the physical state. The main text correctly explains that the P's include recoupling among different quark pairings, but a redistribution of this magnitude (2% at the pole to ≈34% of the norm) should be substantiated, for example by displaying the overlap matrix that underlies Eqs. (52)–(53) or by testing the same overlap factors against an independent observable. I also recommend that the abstract and conclusion state explicitly that the 98%/2% figures are quark–diquark pole weights and not light-front Fock-state probabilities.

Minor objections (7)

[Eq. (36)] The second term of Eq. (36) appears to carry a duplicated normalization factor √Z_N; only one such factor should multiply the axial-vector contribution.
[Various] Typos to correct: 'Rita-Schwinger' should be 'Rarita–Schwinger' (below Eq. (37)); 'diqaurk' (Sec. II); 'follwoing' (Sec. II); 'insted' (Sec. II); 'Laudau' (Table III caption); 'spliting' (Sec. IX); 'quadrplet' (Sec. III).
[App. F, Sec. F3] In the passage introducing the ∆++ wave functions, the state is referred to as 'the proton'; it should be the delta baryon.
[Sec. IX, near Fig. 13] The sentence 'In Fig. 13, we evolve the ¯u and ¯d antiquark helicity PDFs' should read 'transversity PDFs', consistent with the section and the figure caption.
[Sec. VII; Table XII; Figs. 5–6] The two choices α_s(µ0) = 0.87 and 1.78 change the evolved second Mellin moments by 20–30% (⟨x⟩_u: 0.416 vs 0.321 at 2 GeV), so the PDF comparison brackets the lattice and global results rather than predicting them at the few-percent level; the summary of the agreement should say so explicitly.
[Table II] Comparing the ILM quark mass at µ0 ≈ 1/ρ with the FLAG value at 2 GeV conflates scales, since the light-quark mass runs with the renormalization scale; the RG-invariant combination in the last column is the meaningful comparison, and the surrounding sentence should say so.
[Eq. (42)] The light-front momentum wave functions (46)–(47) follow from the model integral of Eq. (42) with the parameter λ_N fixed by normalization rather than from a direct evaluation of the k− integrals of Eq. (41); a sentence stating this explicitly, together with the sensitivity of the results to the functional form of ϕ_N, would help the reader assess the LF continuation.

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The paper

The paper claims

The paper derives the nucleon bound state, its axial charge, and its valence parton distributions from instanton-induced quark-quark correlations, with no baryon-sector free parameters.

The reading

This paper claims that the nucleon, including its mass, its axial and tensor charges, and its leading valence quark content, can be derived from the topological structure of the QCD vacuum. Working in the instanton liquid model with the two vacuum parameters already fixed by the physical pion and rho masses, the authors derive diquark interaction kernels from single instantons and from instanton-anti-instanton molecules, embed them in a covariant, permutation-symmetric three-quark Bethe-Salpeter-Faddeev equation, and obtain a nucleon bound state at $m_N = 939.53$ MeV with a 98% scalar and 2% axial-vector diquark composition. The same covariant amplitude, analytically continued to the light front, yields an axial charge $g_A = 1.265$ and leading-twist unpolarized, helicity, and transversity parton distributions whose QCD-evolved moments are comparable to lattice determinations and global fits. If the paper is right, the same semiclassical vacuum configurations that break chiral symmetry also organize the nucleon's spectrum, spin structure, and partonic content, with confinement and higher Fock components completing rather than replacing that picture.

What carries the argument

The carrying object is the instanton-induced diquark vertex: the 't Hooft interaction transposed and Fierzed into quark-quark channels, supplemented by a molecular instanton-anti-instanton term, with couplings $G_I$ and $G_{IA}$ fixed by the physical pion and rho masses and the instanton size $\rho = 0.33$ fm. These kernels are resummed in bubble chains into diquark propagators, which are pole-approximated with the diquark masses and quark-diquark couplings taken from the two-body calculation. The three-quark Bethe-Salpeter-Faddeev equation is then reduced to a quark-diquark equation; under the static approximation, a momentum-independent quark propagator $-i/M$ inside the exchange kernel, it becomes a diagonalizable gap-like equation whose largest eigenvalue fixes $m_N = 939.53$ MeV and the vertex coefficients $\alpha_1, \alpha_2, \alpha_3$ that enter the light-front wave function. From that wave function the axial and tensor charges and the three families of parton distributions are computed as overlaps, with the inverse instanton size setting the initial scale $\mu_0 \approx 0.63$ GeV for DGLAP evolution to 2 GeV.

Extended reading notes

Core claim

The central claim is that instanton-induced interactions alone, with no baryon-sector free parameters, produce the nucleon bound state and its leading partonic structure. The paper derives the diquark channels from the 't Hooft vertex transposed into quark-quark form, with single instantons generating a strong attraction in the color-antisymmetric isoscalar scalar channel ($m_{qqS} \simeq 592$ MeV) and instanton-anti-instanton molecules supplying the weaker spin-one channels, of which only the axial-vector diquark binds near the two-quark threshold. Solving the covariant, $S_3$-permutation-symmetric Bethe-Salpeter-Faddeev equation in a quark-diquark reduction gives $m_N = 939.53$ MeV with scalar weight 98% and axial-vector weight 2%; omitting the axial-vector component shifts the mass upward by 9.2 MeV. Although the axial-vector weight is small, scalar-axial interference and interference among different quark pairings generate $g_A = 1.265$, against a scalar-only value near unity. The same amplitude, continued to the light front, yields the leading three-quark wave function from which the unpolarized, helicity, and transversity distributions are computed at the intrinsic scale $\mu_0 \simeq 0.63$ GeV and evolved to 2 GeV for comparison with lattice QCD and global fits. The paper also reports no $\Delta$ bound state, which it attributes to the absence of the attractive scalar channel in the $\Delta$ and to the complementary role of confinement.

Load-bearing premise

The load-bearing approximation is the reduction of the covariant three-quark Bethe-Salpeter-Faddeev equation to a diagonalizable quark-diquark gap equation: the diquark propagators are pole-approximated and the quark propagator inside the exchange kernel is replaced by the momentum-independent form $-i/M$, with no full covariant solution shown for comparison. If this simplification shifts the bound-state pole by a few MeV or the vertex coefficients by a few percent, the quoted $m_N = 939.53$ MeV, the 98/2 decomposition, and $g_A = 1.265$ would change by material amounts.

Editorial extensions

If this is right

With the instanton density and size fixed by the meson sector, the baryon calculation runs with no additional free parameters, so a wrong nucleon mass or axial charge would directly test the topological picture rather than a single tuning choice.
A small spin-one admixture can have a large effect on spin observables: the 2% axial-vector weight shifts the nucleon mass by only about 9 MeV but, through scalar-axial and permutation interference, raises $g_A$ from near 1 to 1.265.
One covariant amplitude now serves simultaneously as the Faddeev bound state and the leading light-front Fock wave function, so nucleon charges and collinear parton distributions are tied to the same three-quark structure rather than modeled separately.
The absence of a bound $\Delta$ in this kernel identifies the long-distance binding that confinement must supply, giving a clean division of labor: topology sets the valence correlations, confinement completes the spectrum.
Higher Fock components such as $qqq\bar{q}q$ and $qqqg$ are expected to provide the nonperturbative sea that the leading-Fock calculation lacks, so measured sea-quark flavor asymmetries quantify how much of the nucleon lies beyond the topological valence core.

Reading between the lines

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

The paper's own numbers show that diquark probabilities are basis-dependent: the quark-diquark pole weights ($w_s = 98\%$, $w_a = 2\%$) differ sharply from the fully symmetrized Fock-space norms ($P_{ss} = 0.305$, $P_{sa} = 0.350$, $P_{aa} = 0.339$), so a lattice determination of three-quark overlap probabilities could decide which decomposition is the physical one.
Because the evolved predictions depend strongly on the low-scale coupling, spanning $\alpha_s(\mu_0)$ from 0.87 to 1.78, the model's testable content at 2 GeV is a band rather than a line; a measurement that pins down the intrinsic scale, such as the gluon momentum fraction at low resolution, would collapse that freedom.
The RG-invariant ratio $\delta d/\delta u \approx 0.292$ sits above the lattice values of 0.259 and 0.249, so a more precise lattice determination of the tensor charges would provide a scale-independent test of the instanton-induced spin-flavor structure that no refitting of $\alpha_s$ could absorb.

Original abstract

Instantons, as topologically nontrivial gauge field configurations, provide a microscopic mechanism for chiral symmetry breaking and the $U_A(1)$ anomaly in QCD. Within the instanton liquid model (ILM), instantons generate multi-fermion interactions governed by their density and size. Using this framework, we develop a microscopic description of the nucleon based on instanton-induced quark--quark correlations. We derive the diquark interaction kernels arising from single instantons and instanton--anti-instanton molecules, revealing a strong attraction in the scalar channel and weaker correlations in the spin-one channels. Embedding these interactions in a covariant, permutation-symmetric Bethe--Salpeter--Faddeev equation yields a nucleon mass $m_N=939.53~{\rm MeV}$, with a predominantly scalar-diquark component ($98\%$) and a small axial-vector admixture ($2\%$) around the mass pole. Despite its small weight, the small admixture generates substantial spin--isospin interference, yielding an axial charge $g_A=1.265$, close to experiment. The resulting covariant amplitude is analytically continued to the light front to obtain the leading three-quark wave function. The corresponding unpolarized, helicity, and transversity parton distributions, along with their QCD-evolved moments, provide a direct connection to lattice QCD calculations and global-fit analyses. Our results show that the topology of the QCD vacuum organizes not only the nucleon mass and spin structure, but also its leading valence partonic content, providing a microscopic foundation for a more complete description including confinement and higher Fock components.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 7,212 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.66 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.39 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Instantons, as topologically nontrivial gauge field configurations, provide a microscopic mechanism for chiral symmetry breaking and the $U_A(1)$ anomaly in QCD. Within the instanton liquid model (ILM), instantons generate multi-fermion interactions governed by their density and size. Using this framework, we develop a microscopic description of the nucleon based on instanton-induced quark--quark correlations. We derive the diquark interaction kernels arising from single instantons and instanton--anti-instanton molecules, revealing a strong attraction in the scalar channel and weaker correlations in the spin-one channels. Embedding these interactions in a covariant, permutation-symmetric Bethe--Salpeter--Faddeev equation yields a nucleon mass $m_N=939.53~{\rm MeV}$, with a predominantly scalar-diquark component ($98\%$) and a small axial-vector admixture ($2\%$) around the mass pole. Despite its small weight, the small admixture generates substantial spin--isospin interference, yielding an axial charge $g_A=1.265$, close to experiment. The resulting covariant amplitude is analytically continued to the light front to obtain the leading three-quark wave function. The corresponding unpolarized, helicity, and transversity parton distributions, along with their QCD-evolved moments, provide a direct connection to lattice QCD calculations and global-fit analyses. Our results show that the topology of the QCD vacuum organizes not only the nucleon mass and spin structure, but also its leading valence partonic content, providing a microscopic foundation for a more complete description including confinement and higher Fock components.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Instantons, as topologically nontrivial gauge field configurations, provide a microscopic mechanism for chiral symmetry breaking and the $U_A(1)$ anomaly in QCD.	background 1.00	0.06	0.04	asserted 1.00
s1	Within the instanton liquid model (ILM), instantons generate multi-fermion interactions governed by their density and size.	background 0.90	0.08	0.02	asserted 1.00
s2	Using this framework, we develop a microscopic description of the nucleon based on instanton-induced quark--quark correlations.	contribution 0.86	0.94	0.06	asserted 0.99
s3	We derive the diquark interaction kernels arising from single instantons and instanton--anti-instanton molecules, revealing a strong attraction in the scalar channel and weaker correlations in the spin-one channels.	result 0.72	0.96	0.08	asserted 1.00
s4	Embedding these interactions in a covariant, permutation-symmetric Bethe--Salpeter--Faddeev equation yields a nucleon mass $m_N=939.53~{\rm MeV}$, with a predominantly scalar-diquark component ($98\%$) and a small axial-vector admixture ($2\%$) around the mass pole.	result 1.00	0.96	0.23	asserted 1.00
s5	Despite its small weight, the small admixture generates substantial spin--isospin interference, yielding an axial charge $g_A=1.265$, close to experiment.	result 0.80	0.93	0.09	asserted 1.00
s6	The resulting covariant amplitude is analytically continued to the light front to obtain the leading three-quark wave function.	method 0.98	0.95	0.04	asserted 0.99
s7	The corresponding unpolarized, helicity, and transversity parton distributions, along with their QCD-evolved moments, provide a direct connection to lattice QCD calculations and global-fit analyses.	result 0.45	0.83	0.08	asserted 0.99
s8	Our results show that the topology of the QCD vacuum organizes not only the nucleon mass and spin structure, but also its leading valence partonic content, providing a microscopic foundation for a more complete description including confinement and higher Fock components.	interpretation 0.99	0.93	0.90	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.92.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Feynman diagrams for the fermionic bubble resummation in the diquark channels under the

**Figure 2.** FIG. 2. Baryon bound state in instanton vacuum

**Figure 3.** FIG. 3. The quark exchange matrix in color space

**Figure 4.** FIG. 4. Comparison of the valence

**Figure 7.** FIG. 7. Comparison of the helicity distributions at

**Figure 11.** FIG. 11. Comparison of the

**Figure 13.** FIG. 13. ILM anti-quark transversity distribution

**Figure 15.** FIG. 15. Second Mellin moment of PDFs of

**Figure 16.** FIG. 16. The Feynman diagrams for the quark-diquark bubble resummation in the baryon channels.

Showing 11 of the paper's 16 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.39215 by the authors.

Discussion

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