Nucleon and singly heavy baryons from the QCD instanton vacuum

Hyun-Chul Kim; Yongwoo Choi

REVIEW 3 major objections 6 minor 1 cited by

Nucleon and singly heavy baryons from the QCD instanton vacuum

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A theory built on the QCD instanton vacuum predicts the nucleon and singly heavy baryon mass splittings with no fitted quark mass.

desk verdict A solid, genuinely new instanton-vacuum soliton calculation whose quantitative predictions rest on an untested |F| analytic-continuation choice; worth publishing after a sensitivity analysis. read the letter →

arxiv 2501.12114 v1 pith:7CAENQ7E submitted 2025-01-21 hep-ph hep-lat

Yongwoo Choi , Hyun-CHul Kim This is my paper · ORCID

classification hep-phhep-lat

keywords QCDinstantonvacuumeffectivechiraltheorynucleonsinglyheavybaryonsmomentum-dependentdynamicalquarkmasspionmeanfieldsolitonzero-modequantizationbaryonsplittings

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 builds an effective chiral theory of the nucleon directly from the QCD instanton vacuum, keeping the momentum dependence of the dynamical quark mass. In this picture the nucleon is a bound state of $N_c$ valence quarks held together by a pion mean field that the quarks themselves create; minimizing the classical energy gives $M_{\rm cl}=1.2680$ GeV with the zero-virtuality quark mass $M_0=359$ MeV fixed by the instanton-vacuum gap equation, not by fitting baryons. Zero-mode quantization produces the spin and isospin quantum numbers, and the moments of inertia yield $M_{\Delta-N}=213.67$ MeV for the light sector and $M_{\Sigma_Q-\Lambda_Q}=206.20$ MeV for singly heavy baryons, where the heavy quark enters as a static color source. The point is that one self-consistent framework, with no adjustable quark mass, reproduces both light- and heavy-baryon splittings and preserves the instanton-vacuum content for future gluonic observables.

What carries the argument

The load-bearing object is the nonlocal chiral Dirac operator $D[U] = i\gamma_\mu\partial_\mu + iM_0\, F(i\partial)\, U^{\gamma_5}\, F(i\partial)$, where $F(k\bar\rho)$ is the Fourier transform of the fermionic zero-mode profile in the instanton background and $U^{\gamma_5}$ is the chiral pion field. It supplies the quark form factor that enters the self-consistent pion profile, acts as a natural ultraviolet regulator, and through the residue identity $z_{\rm val} = [1 + i\,\partial E_{\rm val}/\partial\omega]^{-1}$ defines the valence-quark wave-function renormalization that makes the baryon number come out exactly one. The numerical scheme is a Hartree-style iteration: diagonalize the Hamiltonian for a trial profile, feed the eigenstates into the equations of motion, and repeat to the minimum of Eq. (32); the resulting profile is broader than in a local treatment with a constant quark mass. Quantization of the rotational zero modes turns the soliton into a spherical top with moment of inertia $I=I_{\rm val}+I_{\rm sea}$, giving the mass formula $M_{S=T}=M_{\rm cl}+S(S+1)/(2I)$.

What would settle it

Re-run the self-consistent minimization with a different treatment of the timelike form factor, for example using $\mathrm{Re}\,F(k)$ or a dispersion-theoretic continuation, and compare $M_{\rm cl}$, $I$, and the two splittings; if $M_{\Delta-N}$ moves by more than a few tens of MeV, the quoted predictions hinge on the $|F(k)|$ choice.

Watch

Extended reading notes

Core claim

The central claim is that the nonlocal effective action $S_{\rm eff}[U] = -N_c\,\mathrm{Tr}\log D[U]$, with the momentum-dependent quark mass $M(k)=M_0 F(k\bar\rho)^2$ coming from the instanton zero modes, is a working theory of baryons rather than a toy. The classical nucleon mass is the minimum of $N_c$ times the valence-quark level energy plus the Dirac-sea energy, Eq. (32), and with $M_0=359$ MeV the minimization yields $M_{\rm cl}=1.2680$ GeV. The same action, quantized by slow rotation, gives a moment of inertia $I=1.3853$ fm and hence $M_{\Delta-N}=213.67$ MeV; with $N_c-1$ valence quarks and a static heavy quark it gives $M_{\Sigma_Q-\Lambda_Q}=206.20$ MeV. The discovery, as the authors state it, is that the momentum-dependent mass acts as a natural regulator that keeps the chiral anomaly intact and lets the instanton vacuum set the scale, so the splittings are predictions of the vacuum rather than fitted model parameters.

Load-bearing premise

The computation assumes that the quark form factor, which becomes complex for $k^2<0$, can be replaced by its absolute value $|F(k)|$ when constructing the pion mean field; all the reported numbers depend on that substitution, and the paper does not test its sensitivity.

Editorial extensions

If this is right

If the central claim holds, the $\Delta$--$N$ splitting of about 214 MeV and the $\Sigma_Q$--$\Lambda_Q$ splitting of about 206 MeV follow from the instanton vacuum's size and density, not from tuned quark masses.
The momentum-dependent mass removes the need for a separate regularization of the effective action, so the anomalous Wess--Zumino--Witten structure is preserved automatically.
The baryon number of the nucleon is carried entirely by the $N_c$ valence quarks, with the sea-quark contribution vanishing once the gauge connection restores current conservation.
The same machinery describes singly heavy baryons as $N_c-1$ light valence quarks plus a static heavy quark, making the heavy-baryon spectrum a by-product of the light-quark dynamics.
The framework is ready to compute gluonic operators of light and heavy baryons, which is the stated motivation for the electron-ion collider era.

Reading between the lines

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

Editorial inference: the roughly 40% growth in the sea-quark moment of inertia relative to the constant-mass model suggests that nonlocal effects will show up in Dirac-sea-sensitive observables such as axial charges and quark spin fractions; a lattice calculation in the same nonlocal action would test this.
Editorial inference: because the $|F(k)|$ prescription is used only in the timelike region, replacing it by a dispersion-relation continuation and re-minimizing is a natural stress test; the robustness of the two splittings to that choice determines how much of the result is physical.
Editorial inference: applying the framework to the $N_c-2$ valence-quark sector could decide whether doubly heavy baryons can be described without the $M_0\gtrsim 600$ MeV barrier found in the constant-mass model, although the paper regards such a bound state as unlikely.

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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

Short version: this is a competent and genuinely new model calculation, not a breakthrough. The new thing is a self-consistent chiral soliton that keeps the full momentum-dependent quark mass from the instanton vacuum, fixes M0=359 MeV via the gap equation instead of fitting it to baryons, and extends the same machinery to singly heavy baryons. That is worth having.

What it does well: the instanton form factor acts as a natural regulator, so the usual ad hoc regularization of the local chiQSM is avoided; the nonlocal corrections to the moment of inertia are computed explicitly; the baryon-number argument with the gauge connection is careful and gives B=1 from the valence level alone; the lattice comparison for M(k) is encouraging; and the f_pi=90.4 MeV check shows the inputs are sane. The final splittings (Delta-N 213.7 MeV, Sigma_Q-Lambda_Q 206.2 MeV) are in the right ballpark.

The soft spots are real. The most load-bearing is the treatment of the form factor at timelike momenta. After Eq. (35) the paper says it uses |F(k)| to determine the wave function. That choice enters the Hamiltonian, the valence and sea energies, the equations of motion, and the moments of inertia. No test of any alternative continuation (Re F, Im F, principal value) is reported, and no argument shows |F| is unique. Until a sensitivity check is done, the agreement with data cannot be cleanly attributed to the instanton-vacuum dynamics. A second, milder issue is that R_bar=0.98 fm is an input from instanton-vacuum phenomenology, not derived from baryon observables; M0 is therefore not literally parameter-free, just not fitted to baryons. A third is the absence of error estimates or a scan over rho_bar and R_bar. Finally, the experimental agreement is qualitative: the Delta-N result is about 20% below the PDG center (though the Delta width makes the comparison fuzzy), and the charm splitting is ~23% above data; bottom looks better, as expected in the infinite-heavy-quark limit.

Overall: the central argument holds up, but the quantitative claims are conditional on the |F| prescription. The paper deserves a serious referee, and I would send it to review with the request that the authors quantify the sensitivity to the timelike continuation and release at least the self-consistent numerics. With that, it becomes a useful framework for EIC-oriented gluonic observables.

Referee Report

3 major / 6 minor

Summary. The paper constructs a nonlocal effective chiral theory for the nucleon and singly heavy baryons starting from the low-energy QCD partition function of the instanton vacuum, retaining the full momentum dependence of the dynamical quark mass M(k)=M0 F(k)^2. The value M0=359 MeV is fixed by the gap equation from the instanton-vacuum inputs rho_bar=1/3 fm and R_bar=0.98 fm, and the pion decay constant f_pi=90.4 MeV emerges as a cross-check. The nucleon is described as N_c valence quarks bound by a self-consistent hedgehog pion mean field; the classical mass, the moment of inertia, and the Delta-N and Sigma_Q-Lambda_Q mass splittings are computed, yielding 1268 MeV, 1.3853 fm, 213.67 MeV, and 206.20 MeV, respectively. The central claim is that these baryon mass splittings are predictions of the instanton-vacuum dynamics without fitting to baryon observables.

Significance. If the results are robust, this work is a valuable step: it connects instanton-vacuum parameters to baryon mass splittings within a single framework and offers a path toward gluonic observables at the EIC. The paper has clear strengths: the derivation from the instanton-vacuum partition function is explicit, M0 is not fitted to baryon data, the baryon number is explicitly shown to be carried by the valence quarks, and the comparison with lattice M(k) and with f_pi provides independent cross-checks. The main quantitative predictions, however, rest on an untested analytic-continuation prescription for the quark form factor and on a truncated Taylor expansion, which currently leaves the central claim less secure than the authors assert.

major comments (3)

[Section III, after Eq. (35)] The choice to replace the timelike form factor F(k) by |F(k)| is an uncontrolled modeling assumption. In the self-consistent calculation the quark virtuality is k^2 = -E_val^2 < 0, where F(k) is complex because of the Bessel-function branch point. The text states 'we assume that the absolute value of the quark form factor, |F(k)|, is used to determine the wave function,' and this prescription enters the Dirac Hamiltonian (Eq. 23), the valence and sea energies (Eqs. 26-34), the equations of motion (Eqs. 33-34), and the moments of inertia (Eqs. 50-54). No sensitivity test is reported, and no argument is given that |F| is the natural continuation; Re F, Im F, or a principal-value prescription are equally well defined from the expressions in Eq. (35). Moreover, |F| is non-analytic, which puts into question the residue-theorem derivation of Eq. (26). Without a sensitivity analysis or a derivation of the correct continuation, the agreement with experiment cannot be attributed uniquely to the instanton-vacuum dynamics.
[Section V, Eq. (48)] The rotational corrections are obtained by Taylor expanding F(i partial) around k^2=0 and keeping only first and second derivatives, but the relevant quark poles in the self-consistent solution are at k^2 = -E_val^2, where the expansion point is not obviously within the radius of convergence. The operators t^a and T^{ab} in Eq. (50), and hence the moments of inertia in Eqs. (52) and (54), depend on F_4 and F_44. The paper itself notes that dropping the nonlocal derivative terms changes the total moment of inertia from 1.385 fm to 2.065 fm, so the splitting predictions are sensitive to this expansion. Please test the stability of I and of the mass splittings under including higher-order terms or using the full nonlocal operator.
[Section VI, Eq. (79)] The mass formula for singly heavy baryons, M_B = M_cl + m_Q + (1/(2 Itilde)) S(S+1), uses the total baryon spin S in the rotational energy, but the rotational Hamiltonian for the N_c-1 soliton should be governed by the light-quark cluster spin S' as defined in Eq. (78). With the printed formula, the Sigma_Q-Lambda_Q splitting would not equal 1/Itilde, and the quoted value of 206.20 MeV cannot be reproduced; the numerical result corresponds to using S'=1 for the Sigma and S'=0 for the Lambda. This is a load-bearing inconsistency that must be corrected, either by writing S'(S'+1) in Eq. (79) or by clarifying the notation for S throughout Section VI.

minor comments (6)

[Eq. (35)] The expression for Re[F(k)] contains a term i I1(z) Y1(-iz) inside a supposedly real part; please check the formula for typographical errors.
[Section V] The text quotes the experimental Delta-N mass splitting as 267.62 MeV from Ref. [76], but the PDG Breit-Wigner masses give about 293 MeV; please clarify the source of 267.62 MeV and the stated range of 200-400 MeV.
[Section VI] The text writes 'M exp. Lambda_c = 2.286 MeV' and 'M exp. Lambda_b = 5.619, respectively'; the first should be GeV and the second is missing units (GeV).
[Throughout] There are several typos: 'constributions' in Section II, 'pseusoscalar' in Eq. (77), 'soluion' in Section VI, and 'isospinglet' in Section VI; 'Nöther current' should be 'Noether current' in the discussion after Eq. (42).
[Table IV] The column heading 'IT' should be 'I' or 'Itilde' to match the notation in Eq. (80) and the surrounding text.
[Section VI] The notation S in Eq. (79) is inconsistent with the definition of S and S' in Eq. (78); please make the spin notation uniform throughout the section.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the gap-equation input M0 and the computed baryon splittings are not linked by construction.

full rationale

The paper's central outputs, Mcl = 1.2680 GeV, I = 1.3853 fm, MDelta-N = 213.67 MeV, and MSigmaQ-LambdaQ = 206.20 MeV, are not fitted to the experimental baryon spectrum. The dynamical quark mass at zero virtuality, M0 = 359 MeV, is determined by the gap equation from the instanton-vacuum inputs rho_bar = 1/3 fm and R_bar = 0.98 fm, which are fixed by independent vacuum phenomenology such as the gluon condensate and topological susceptibility, not by the baryon masses. The pion decay constant fpi = 90.4 MeV is presented as a cross-check, and the momentum dependence of M(k) is compared with lattice data, providing external anchoring. The self-consistent profile is obtained by minimizing the classical mass in Eq. (32), and the moments of inertia are computed from the eigenstates of the Dirac Hamiltonian via Eqs. (52) and (54); the mass-splitting formulas Eq. (70) and Eq. (79) are standard collective-quantization identities evaluated with these computed values. The assumption that |F(k)| is used for the wave function at timelike momenta, stated after Eq. (35), is an explicit modeling choice rather than a fitted input, and while it affects the numerics and deserves a sensitivity study, it does not make the predictions equivalent to the inputs by construction. The heavy-baryon formalism is imported from prior published work by the authors, but that work has independent content and has been tested against other observables; it is not invoked as an unverified uniqueness theorem. The identity B_val = 1 in Eq. (41) follows from the definition of zval as the residue prefactor and serves as a consistency check, not as a fitted prediction. No load-bearing step reduces to its own input, so the paper is not circular.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central outputs depend on two instanton-liquid inputs (rho_bar, R_bar) and a set of structural approximations. M0 is derived from the gap equation rather than fitted to baryon spectra. The most ad hoc element is the |F(k)| prescription for timelike momenta, which directly shapes the self-consistent profile and hence the predicted splittings. The heavy-quark static limit is a standard approximation but is known to be better for bottom than for charm baryons.

free parameters (2)

Average instanton size rho_bar = 1/3 fm
Standard input from instanton liquid phenomenology; defines the quark form factor F(k) and enters the gap equation; not fitted to baryon masses in this paper. Cited in Section II.
Average instanton interdistance R_bar = 0.98 fm
Sets the instanton density N/V in the gap equation, fixing M0 = 359 MeV. The value 0.98 fm is quoted in Section III and differs from the approximate 1 fm in Section II; its derivation is not shown in this paper.

assumptions (7)

domain assumption Large Nc limit and saddle-point approximation for the pion field
The nucleon is treated as Nc valence quarks in a self-consistent pion mean field, with pion fluctuations suppressed as 1/Nc. Invoked in Section III and throughout.
domain assumption Chiral limit (mu = md = 0)
The pion mass is set to zero; this affects the energies Eval and Esea and the profile. Stated in Section II.
domain assumption Hedgehog symmetry for the chiral field
The pion mean field is restricted to the hedgehog form U = exp[i(tau dot n) Theta(r)] in Eq. (24), reducing the problem to a single profile Theta(r). Standard in chiral soliton models.
domain assumption Dilute instanton liquid approximation
The instanton ensemble is treated in the dilute liquid approximation with average size and spacing; the form factor F(k) is taken from single-instanton fermionic zero modes. From Refs. [36-40].
ad hoc to paper Use of |F(k)| for timelike momenta (k^2 < 0)
The complex form factor on the timelike side is replaced by its absolute value for determining wavefunctions. Stated after Eq. (35) in Section III. Load-bearing for the self-consistent profile.
ad hoc to paper Taylor expansion of F(i d) around k^2 = 0 for rotational corrections
Section V Eq. (48) expands the non-analytic form factor around k^2 = 0 to O(Omega^2). No convergence argument is given, and these terms reduce Ival by about 40%.
domain assumption Heavy-quark static limit (mQ -> infinity)
The heavy quark is treated as a static color source with no interaction with the pion field at leading order. Used in Section VI; 1/mQ corrections are postponed.

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Pith. "Pith review of Nucleon and singly heavy baryons from the QCD instanton vacuum." pith.science (2026). https://pith.science/paper/7CAENQ7E

@misc{pith2026250112114,
  author       = {Pith},
  title        = {Pith review of: Nucleon and singly heavy baryons from the QCD instanton vacuum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CAENQ7E}},
  note         = {Machine review of arXiv:2501.12114}
}

abstract

We construct an effective chiral theory for the nucleon, based on the low-energy effective QCD partition function from the QCD instanton vacuum. We fully consider the momentum-dependent dynamical quark mass whose value at the zero virtuality of the quark is determined by the gap equation from the instanton vacuum, $M_0=359$ MeV. The nucleon emerges as a state of $N_c$ valence quarks bound by the pion mean field, which was created self-consistently by the $N_c$ valence quarks. In the large Euclidean time, the classical nucleon mass is evaluated by minimizing the sum of the $N_c$ discrete-level energies and the Dirac-continuum energy: $M_{\text{cl}}=1.2680$ GeV. The pion mean-field solution turns out broader than the local chiral quark-soliton model. The zero-mode quantization furnishes the nucleon with proper quantum numbers such as the spin and isospin. We compute the moment of inertia $I=1.3853$ fm by using the self-consistent mean-field solution, which yields the $\Delta -N$ mass splitting $M_{\Delta-N} =213.67$ MeV. In the same manner, singly heavy baryons can be described as a bound state of the $N_c-1$ valence quarks with the corresponding pion mean field, with the heavy quark regarded as a static color source. The mass splitting of the singly heavy baryons is obtained to be $M_{\Sigma_Q-\Lambda_Q}=206.20$ MeV, which are in good agreement with the experimental data. The effective chiral theory developed in the present work will provide a solid theoretical framework to investigate gluonic observables of both the light and singly heavy baryons.

Figures

Figures reproduced from arXiv: 2501.12114 by the authors.

**Figure 2.** FIG. 2. Nucleon correlation function consisting of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. The self-consistent profile function Θ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. The self-consistent profile function [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

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Works this paper leans on

93 extracted references · 35 canonical work pages · cited by 1 Pith paper

[1]

Masson, Phil

O. Masson, Phil. Mag. 41, 281 (1921)

1921
[2]

Frisch and O

R. Frisch and O. Stern, Zeit. f. Phys. 85, 17 (1933)

1933
[3]

project the Nc valence-quark states to the corresponding baryon states with definite quantum numbers given. Noting that the matrix element of the evolution operator can be expressed as Z R(T /2) R(−T /2) DR(t) exp " − 1 2I Z T /2 −T /2 dt S2 a # = R(T /2) exp − 1 2I S2 a R(−T /2) = X SS3T3 ψ(T T3)(SS3)(R(T /2))ψ∗ (T T3)(SS3)(R(−T /2)) exp − 1 2I S(S + 1) ...

1920
[4]

Estermann and O

I. Estermann and O. Stern, Zeit. f. Phys. 85, 4 (1933)

1933
[5]

Estermann, O

I. Estermann, O. C. Simpson, and O. Stern, Phys. Rev. 52, 535 (1937)

1937
[6]

L. W. Alvarez and F. Bloch, Phys. Rev. 57, 111 (1940)

1940
[7]

Hofstadter, Rev

R. Hofstadter, Rev. Mod. Phys. 28, 214 (1956)

1956
[8]

D. R. Yennie, M. M. L´ evy, and D. G. Ravenhall, Rev. Mod. Phys. 29, 144 (1957)

1957

Show all 93 references

[9]

Breidenbach, J

M. Breidenbach, J. I. Friedman, H. W. Kendall, E. D. Bloom, D. H. Coward, H. C. DeStaebler, J. Drees, L. W. Mo, and R. E. Taylor, Phys. Rev. Lett. 23, 935 (1969)

1969
[10]

J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975 (1969)

1969
[11]

Fritzsch, M

H. Fritzsch, M. Gell-Mann, and H. Leutwyler, Phys. Lett. B 47, 365 (1973)

1973
[12]

Gross et al., Eur

F. Gross et al., Eur. Phys. J. C 83, 1125 (2023), arXiv:2212.11107 [hep-ph]

2023 arXiv
[13]

De Rujula, H

A. De Rujula, H. Georgi, and S. L. Glashow, Phys. Rev. D 12, 147 (1975)

1975
[14]

Chodos, R

A. Chodos, R. L. Jaffe, K. Johnson, and C. B. Thorn, Phys. Rev. D 10, 2599 (1974)

1974
[15]

T. H. R. Skyrme, Proc. Roy. Soc. Lond. A 260, 127 (1961)

1961
[16]

G. S. Adkins, C. R. Nappi, and E. Witten, Nucl. Phys. B 228, 552 (1983)

1983
[17]

A. W. Thomas, Adv. Nucl. Phys. 13, 1 (1984)

1984
[18]

L. Y. Glozman, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998), arXiv:hep-ph/9706507

1998 arXiv
[19]

Weigel, R

H. Weigel, R. Alkofer, and H. Reinhardt, Nucl. Phys. B 387, 638 (1992)

1992
[20]

Alkofer, H

R. Alkofer, H. Reinhardt, and H. Weigel, Phys. Rept. 265, 139 (1996), arXiv:hep-ph/9501213

1996 arXiv
[21]

Golli, W

B. Golli, W. Broniowski, and G. Ripka, Phys. Lett. B 437, 24 (1998), arXiv:hep-ph/9807261

1998 arXiv
[22]

Broniowski, B

W. Broniowski, B. Golli, and G. Ripka, Nucl. Phys. A 703, 667 (2002), arXiv:hep-ph/0107139

2002 arXiv
[23]

Eichmann, R

G. Eichmann, R. Alkofer, A. Krassnigg, and D. Nicmorus, Phys. Rev. Lett. 104, 201601 (2010), arXiv:0912.2246 [hep-ph]

2010 arXiv
[24]

M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979)

1979
[25]

M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 147, 448 (1979)

1979
[26]

B. L. Ioffe and A. V. Smilga, Nucl. Phys. B 232, 109 (1984)

1984
[27]

I. D. King and C. T. Sachrajda, Nucl. Phys. B 279, 785 (1987)

1987
[28]

D. B. Leinweber, A. W. Thomas, and R. D. Young, Phys. Rev. Lett. 92, 242002 (2004), arXiv:hep-lat/0302020

2004 arXiv
[29]

Ali Khan et al

A. Ali Khan et al. (QCDSF-UKQCD), Nucl. Phys. B 689, 175 (2004), arXiv:hep-lat/0312030

2004 arXiv
[30]

Alexandrou, M

C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, A. Vaquero Avil´ es-Casco, and C. Wiese, Phys. Rev. Lett. 119, 142002 (2017), arXiv:1706.02973 [hep-lat]

2017 arXiv
[31]

M¨ uller, D

D. M¨ uller, D. Robaschik, B. Geyer, F. M. Dittes, and J. Hoˇ rejˇ si, Fortsch. Phys.42, 101 (1994), arXiv:hep-ph/9812448

1994 arXiv
[32]

Ji, Phys

X.-D. Ji, Phys. Rev. D 55, 7114 (1997), arXiv:hep-ph/9609381

1997 arXiv
[33]

A. V. Radyushkin, Phys. Lett. B 380, 417 (1996), arXiv:hep-ph/9604317

1996 arXiv
[34]

M. V. Polyakov and P. Schweitzer, PoS SPIN2018, 066 (2019), arXiv:1812.06143 [hep-ph]

2019 arXiv
[35]

M. V. Polyakov and C. Weiss, Phys. Rev. D 60, 114017 (1999), arXiv:hep-ph/9902451

1999 arXiv
[36]

Abdul Khalek et al., Nucl

R. Abdul Khalek et al., Nucl. Phys. A 1026, 122447 (2022), arXiv:2103.05419 [physics.ins-det]

2022 arXiv
[37]

E. V. Shuryak, Nucl. Phys. B 203, 93 (1982)

1982
[38]

Diakonov and V

D. Diakonov and V. Y. Petrov, Nucl. Phys. B 245, 259 (1984)

1984
[39]

Diakonov and V

D. Diakonov and V. Y. Petrov, Nucl. Phys. B 272, 457 (1986)

1986
[40]

Sch¨ afer and E

T. Sch¨ afer and E. V. Shuryak, Rev. Mod. Phys.70, 323 (1998), arXiv:hep-ph/9610451

1998 arXiv
[41]

Diakonov, Prog

D. Diakonov, Prog. Part. Nucl. Phys. 51, 173 (2003), arXiv:hep-ph/0212026

2003 arXiv
[42]

Diakonov, M

D. Diakonov, M. V. Polyakov, and C. Weiss, Nucl. Phys. B 461, 539 (1996), arXiv:hep-ph/9510232

1996 arXiv
[43]

Diakonov, V

D. Diakonov, V. Y. Petrov, and P. V. Pobylitsa, Nucl. Phys. B 306, 809 (1988)

1988
[44]

Blotz, D

A. Blotz, D. Diakonov, K. Goeke, N. W. Park, V. Petrov, and P. V. Pobylitsa, Nucl. Phys. A 555, 765 (1993)

1993
[45]

C. V. Christov, A. Blotz, H.-C. Kim, P. Pobylitsa, T. Watabe, T. Meissner, E. Ruiz Arriola, and K. Goeke, Prog. Part. Nucl. Phys. 37, 91 (1996), arXiv:hep-ph/9604441

1996 arXiv
[46]

Diakonov, in Advanced Summer School on Nonperturbative Quantum Field Physics (1997) pp

D. Diakonov, in Advanced Summer School on Nonperturbative Quantum Field Physics (1997) pp. 1–55, arXiv:hep- ph/9802298. 20

1997
[47]

Witten, Nucl

E. Witten, Nucl. Phys. B 160, 57 (1979)

1979
[48]

Witten, Nucl

E. Witten, Nucl. Phys. B 223, 422 (1983)

1983
[49]

Witten, Nucl

E. Witten, Nucl. Phys. B 223, 433 (1983)

1983
[50]

R. D. Ball and G. Ripka, in Conference on Many-Body Physics(1993) pp. 129–140, arXiv:hep-ph/9312260

1993 arXiv
[51]

Balla, M

J. Balla, M. V. Polyakov, and C. Weiss, Nucl. Phys. B 510, 327 (1998), arXiv:hep-ph/9707515

1998 arXiv
[52]

Diakonov and V

D. Diakonov and V. Y. Petrov, Phys. Lett. B 147, 351 (1984)

1984
[53]

H.-C. Kim, M. M. Musakhanov, and M. Siddikov, Phys. Lett. B 633, 701 (2006), arXiv:hep-ph/0508211

2006 arXiv
[54]

’t Hooft, Phys

G. ’t Hooft, Phys. Rev. D 14, 3432 (1976), [Erratum: Phys.Rev.D 18, 2199 (1978)]

1976
[55]

Choi and H.-C

H.-A. Choi and H.-C. Kim, Phys. Rev. D 69, 054004 (2004), arXiv:hep-ph/0308171

2004 arXiv
[56]

Pagels and S

H. Pagels and S. Stokar, Phys. Rev. D 20, 2947 (1979)

1979
[57]

Holdom, J

B. Holdom, J. Terning, and K. Verbeek, Phys. Lett. B 245, 612 (1990)

1990
[58]

R. D. Bowler and M. C. Birse, Nucl. Phys. A 582, 655 (1995), arXiv:hep-ph/9407336

1995 arXiv
[59]

J. F. Donoghue, E. Golowich, and B. R. Holstein, Dynamics of the Standard Model: Second edition(Cambridge University Press, 2022)

2022
[60]

Gasser and H

J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984)

1984
[61]

Gasser and H

J. Gasser and H. Leutwyler, Nucl. Phys. B 250, 465 (1985)

1985
[62]

Pauli and S

W. Pauli and S. M. Dancoff, Phys. Rev. 62, 85 (1942)

1942
[63]

Diakonov and V

D. Diakonov and V. Y. Petrov, Sov. Phys. JETP 62, 204 (1985)

1985
[64]

Chretien and R

M. Chretien and R. E. Peierls, Proc. Roy. Soc. Lond. A 223, 468 (1954)

1954
[65]

Holdom, J

B. Holdom, J. Terning, and K. Verbeek, Phys. Lett. B 232, 351 (1989)

1989
[66]

Holdom, Phys

B. Holdom, Phys. Rev. D 45, 2534 (1992)

1992
[67]

J. W. Bos, J. H. Koch, and H. W. L. Naus, Phys. Rev. C 44, 485 (1991)

1991
[68]

Broniowski, in Mini-Workshop Bled 1999: Hadrons as Solitons(1999) pp

W. Broniowski, in Mini-Workshop Bled 1999: Hadrons as Solitons(1999) pp. 17–26, arXiv:hep-ph/9909438

1999 arXiv
[69]

P. V. Pobylitsa, Phys. Lett. B 226, 387 (1989)

1989
[70]

Diakonov and V

D. Diakonov and V. Y. Petrov, Sov. Phys. JETP 62, 431 (1985)

1985
[71]

M. M. Musakhanov and H.-C. Kim, Phys. Lett. B 572, 181 (2003), arXiv:hep-ph/0206233

2003 arXiv
[72]

Diakonov and M

D. Diakonov and M. I. Eides, JETP Lett. 38, 433 (1983)

1983
[73]

Jain and S

S. Jain and S. R. Wadia, Nucl. Phys. B 258, 713 (1985)

1985
[74]

P. O. Mazur, M. A. Nowak, and M. Praszalowicz, Phys. Lett. B 147, 137 (1984)

1984
[75]

L. D. Landau and L. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory: Third edition(Butterworth-Heinemann, 1981)

1981
[76]

Kim, H.-Y

J.-Y. Kim, H.-Y. Won, H.-C. Kim, and C. Weiss, Phys. Rev. D 110, 054026 (2024), arXiv:2403.07186 [hep-ph]

2024 arXiv
[77]

Navas et al

S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)

2024
[78]

Yang, H.-C

G.-S. Yang, H.-C. Kim, M. V. Polyakov, and M. Prasza lowicz, Phys. Rev. D94, 071502 (2016), arXiv:1607.07089 [hep-ph]

2016 arXiv
[79]

Isgur and M

N. Isgur and M. B. Wise, Phys. Lett. B 232, 113 (1989)

1989
[80]

Isgur and M

N. Isgur and M. B. Wise, Phys. Rev. Lett. 66, 1130 (1991)

1991
[81]

Georgi, Phys

H. Georgi, Phys. Lett. B 240, 447 (1990)

1990
[82]

Yang and H.-C

G.-S. Yang and H.-C. Kim, Phys. Lett. B 781, 601 (2018), arXiv:1802.05416 [hep-ph]

2018 arXiv
[83]

Kim, H.-C

J.-Y. Kim, H.-C. Kim, and G.-S. Yang, Phys. Rev. D 98, 054004 (2018), arXiv:1801.09405 [hep-ph]

2018 arXiv
[84]

Kim and H.-C

J.-Y. Kim and H.-C. Kim, PTEP 2020, 043D03 (2020), arXiv:1909.00123 [hep-ph]

2020 arXiv
[85]

Yang and H.-C

G.-S. Yang and H.-C. Kim, Phys. Lett. B 801, 135142 (2020), arXiv:1909.03156 [hep-ph]

2020 arXiv
[86]

Kim and H.-C

J.-Y. Kim and H.-C. Kim, Phys. Rev. D 97, 114009 (2018), arXiv:1803.04069 [hep-ph]

2018 arXiv
[87]

Kim and H.-C

J.-Y. Kim and H.-C. Kim, PTEP 2021, 023D02 (2021), arXiv:2008.08296 [hep-ph]

2021 arXiv
[88]

Kim and H.-C

J.-Y. Kim and H.-C. Kim, PTEP 2021, 063D03 (2021), arXiv:1912.01437 [hep-ph]

2021 arXiv
[89]

Suh and H.-C

J.-M. Suh and H.-C. Kim, Phys. Rev. D 106, 094028 (2022), arXiv:2204.13982 [hep-ph]

2022 arXiv
[90]

Suh, J.-Y

J.-M. Suh, J.-Y. Kim, G.-S. Yang, and H.-C. Kim, Phys. Rev. D 106, 054032 (2022), arXiv:2208.04447 [hep-ph]

2022 arXiv
[91]

H.-C. Kim, J. Korean Phys. Soc. 73, 165 (2018), arXiv:1804.04393 [hep-ph]

2018 arXiv
[92]

Kim, H.-C

J.-Y. Kim, H.-C. Kim, G.-S. Yang, and M. Oka, Phys. Rev. D 103, 074025 (2021), arXiv:2101.10653 [hep-ph]

2021 arXiv
[93]

M. A. Shifman, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 95): QCD and Beyond (1995) pp. 409–514, arXiv:hep-ph/9510377

1995 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Masson, Phil

O. Masson, Phil. Mag. 41, 281 (1921)

1921

[2] [2]

Frisch and O

R. Frisch and O. Stern, Zeit. f. Phys. 85, 17 (1933)

1933

[3] [3]

project the Nc valence-quark states to the corresponding baryon states with definite quantum numbers given. Noting that the matrix element of the evolution operator can be expressed as Z R(T /2) R(−T /2) DR(t) exp " − 1 2I Z T /2 −T /2 dt S2 a # = R(T /2) exp − 1 2I S2 a R(−T /2) = X SS3T3 ψ(T T3)(SS3)(R(T /2))ψ∗ (T T3)(SS3)(R(−T /2)) exp − 1 2I S(S + 1) ...

1920

[4] [4]

Estermann and O

I. Estermann and O. Stern, Zeit. f. Phys. 85, 4 (1933)

1933

[5] [5]

Estermann, O

I. Estermann, O. C. Simpson, and O. Stern, Phys. Rev. 52, 535 (1937)

1937

[6] [6]

L. W. Alvarez and F. Bloch, Phys. Rev. 57, 111 (1940)

1940

[7] [7]

Hofstadter, Rev

R. Hofstadter, Rev. Mod. Phys. 28, 214 (1956)

1956

[8] [8]

D. R. Yennie, M. M. L´ evy, and D. G. Ravenhall, Rev. Mod. Phys. 29, 144 (1957)

1957