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Constituent Quark Model and nucleon-Nucleon Potentials

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The constituent quark model reproduces the operator structure of nucleon-meson vertices.

desk verdict A serious, systematic attempt to derive NN vertices from constituent quarks, but the central cancellations are reverse-engineered and one key Gaussian condition has a factor-of-2 error. read the letter →

arxiv 2412.19858 v1 pith:KGS35PFZ submitted 2024-12-25 nucl-th hep-ph

Th. A. Rijken This is my paper · ORCID
classification nucl-thhep-ph PACS 13.75.Cs12.39.Pn21.30.+y
keywords constituentquarkmodelnucleon-nucleonpotentialmesonexchangePaulispinorinvariantsaxial-vectorcouplingspin-orbitinteractionextended-soft-coreorbitalangularmomentum
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

These notes aim to show that the constituent quark model, with quark mass set to one third of the nucleon mass, can produce the same ratios of central, spin-spin, tensor, spin-orbit, and quadratic-spin-orbit Pauli invariants as the phenomenological meson-nucleon-nucleon vertices used in nuclear potentials. The argument works by defining quark-quark-meson interactions and folding them with a Gaussian SU(6) three-quark wavefunction. To make the folded vertex match the nucleon-level vertex, the paper adds extra quark-level couplings for scalar, vector, and axial-vector mesons without new free parameters, plus a Gaussian momentum correlation that removes spurious terms. If correct, this would give a quark-level representation of the nuclear force vertex structure and connect it to the extended-soft-core interaction models used in nuclear and neutron-star physics.

What carries the argument

The object doing the work is the folded quark-quark-meson vertex: a Gaussian SU(6) three-quark wavefunction with nucleon radius parameter $R_N$ is integrated against a quark-level meson vertex, and the result is expanded in Pauli-spinor invariants. The comparison identity is that, after the CQM replacement $m_Q = \sqrt{M'M}/3$, the coefficients of the central, spin-spin, tensor, spin-orbit, and quadratic-spin-orbit operators must equal those of the phenomenological nucleon-level vertex. Extra quark-level derivative couplings and a Gaussian momentum distribution for the exchanged momentum $K$ are the mechanisms that cancel the unwanted $k^2/(M'M)$ and $1/(m_Q^2 R_N^2)$ terms, leaving the desired vertex ratios.

What would settle it

Compute the quark-level scalar, vector, and axial-vector three-point form factors at quark mass $m_Q = M_N/3$ and check the predicted ratios $g_2/g_1 = -8/9$, $f'_{1v}/F_{1v} = 4/9$, $f'_{2v}/F_{2v} = 4/9$, and $g''_a = -3g_a/(8(MR_N)^2)$; a form-factor calculation that does not show these ratios, or a measurement of NN scattering that conflicts with the resulting Pauli-invariant ratios, would falsify the central claim.

Watch

Extended reading notes

Core claim

The central claim is that the constituent quark model with $m_Q = M_N/3$ reproduces, up to order $1/(M'M)$, the Pauli-invariant structure of the nucleon-meson vertices for pseudoscalar, scalar, vector, and axial-vector exchange. The match is achieved by supplementing the standard quark couplings with extra interactions whose ratios are fixed numbers, such as $g_2/g_1 = -8/9$ for scalar mesons, $f'_{1v}/F_{1v} = 4/9$ and $f'_{2v}/F_{2v} = 4/9$ for vector mesons, and an axial-vector coupling $g''_a = -3g_a/(8(MR_N)^2)$ interpreted as the quark orbital angular momentum contribution to the nucleon spin. A momentum correlation between the active quark and the spectator pair, implemented as a Gaussian distribution, eliminates the spurious $1/R_N^2$ terms that would otherwise break the claimed ratio equality. The paper concludes that the conjecture that the ratios of spin-spin, tensor, spin-orbit, and central operators are independent of the nucleon's internal structure is realized in the constituent quark model.

Load-bearing premise

The construction leans on specific extra quark-level interactions and a Gaussian momentum correlation whose parameters are chosen so that the folded vertex equals the phenomenological one; if those ingredients have no independent justification, the match is an exercise in reverse engineering rather than a derivation.

Editorial extensions

If this is right

  • The ratios among central, spin-spin, tensor, spin-orbit, and quadratic-spin-orbit terms in the NN potential are fixed by quark-level couplings with $m_Q = M_N/3$, so the CQM constrains the relative strengths of meson-exchange components.
  • Scalar and vector exchange must be treated together: the extra scalar coupling $g_2/g_1 = -8/9$ and the vector couplings $f'_{1v}/F_{1v} = 4/9$, $f'_{2v}/F_{2v} = 4/9$ cancel the $k^2$ terms that each vertex produces separately.
  • The axial-vector vertex requires an extra coupling tied to quark orbital angular momentum, linking the nucleon spin crisis to the axial-vector meson part of the nuclear force.
  • The Gaussian momentum correlation that removes spurious $1/R_N^2$ terms makes the CQM vertex equivalent to the phenomenological vertex to order $1/(M'M)$, and the same construction can be extended to other baryons.
  • The resulting quark-level interactions connect directly to the extended-soft-core meson-exchange potentials, providing a working input for mixed quark-nuclear matter calculations in neutron stars.

Reading between the lines

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

  • A testable prediction left implicit in the notes is that the ratio conditions ($g_2/g_1=-8/9$, $f'_{1v}/F_{1v}=4/9$, $f'_{2v}/F_{2v}=4/9$, and $g''_a=-3g_a/(8(MR_N)^2)$) should emerge from any quark-level Lagrangian with the same derivative structure; lattice or Dyson-Schwinger three-point functions at $m_Q\simeq M_N/3$ could check them directly.
  • The Gaussian momentum correlation with $\alpha = (3/8)R_N^2$ and $\beta = (3/4)R_N^2$ is effectively a model of the internal momentum distribution of the active quark, and could be confronted with light-front wave functions or generalized parton distributions.
  • If the derivation holds, the same folding procedure would reduce the number of independent coupling constants in hyperon-nucleon and hyperon-hyperon extended-soft-core potentials, since the quark-level ratios would fix the baryon-level vertex ratios across the SU(3) octet.
  • The paper's added couplings are chosen to match the phenomenological vertex; an independent derivation of those couplings from a chiral quark-meson Lagrangian would turn the demonstrated consistency into a full explanation rather than a parameterization.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper aims to derive the meson-nucleon-nucleon (NNM) vertex structure of the extended-soft-core (ESC) potentials from meson exchange between constituent quarks. Quark-meson vertices are folded with the Gaussian SU(6) three-quark wavefunction of Le Yaouanc et al., with quark mass m_Q = M_N/3, and the resulting Pauli-spinor invariants are compared with the phenomenological NNM expansion. The pseudoscalar case is shown to match directly. For scalar, vector, and axial-vector mesons, the author introduces additional derivative couplings at the quark level, a Gaussian momentum distribution in the exchanged momentum K, and (for axial mesons) couplings interpreted as quark orbital angular momentum. The stated conclusion is that the CQM reproduces the ratios of central, spin-spin, tensor, spin-orbit, and quadratic-spin-orbit invariants of the ESC vertices up to order 1/(M'M).

Significance. If the construction were genuinely predictive, it would provide a quark-level foundation for the ESC nucleon-nucleon vertex structure and would connect the CQM to modern NN potentials and neutron-star applications. The Pauli-reduction algebra is set out carefully in the text and appendices, and the pseudoscalar matching in Sec. V is a clean, nontrivial success. The paper also makes the useful observation that the required quark-level ingredients include derivative couplings and a quark-spectator momentum correlation. However, the significance is substantially limited by the fact that the strengths of the additional interactions are fixed by the very NNM coefficients they are meant to explain, so the central claim is currently a consistency construction rather than a derivation. There is also a concrete algebraic inconsistency in the key K-integral condition that must be resolved before the scalar and vector matches are credible.

major comments (3)
  1. [Secs. IV, VI, VII (Eqs. (4.11), (6.6), (6.16), (7.14), (7.23))] The central claim that the CQM 'produces' the empirical Pauli-invariant ratios is weakened by the fact that the additional quark-level couplings are fixed by the NNM coefficients they are meant to reproduce. Eq. (4.11) sets g2/g1 = -8/9 to cancel the extra k^2/(16M'M) term in Eq. (4.8); Eqs. (6.6) and (6.16) set f'_1v/F1v = f'_2v/F2v = 4/9 for the same reason; and Eq. (7.23) fixes g''_a = -3g_a/(8(MR_N)^2) to obtain the required spin-orbit term. The K-distribution parameters in Eq. (4.18) are likewise chosen so that the integrals K1, K2, K3 satisfy the target conditions (4.17). Since no independent derivation or observable is given for these values, the manuscript demonstrates consistency by construction rather than a derivation. The abstract and conclusions should either be rephrased to state this explicitly, or the extra couplings should be derived from an independent physical principle.
  2. [Sec. IV C, Eqs. (4.15)-(4.18), (4.20)-(4.21)] The key condition (4.17)(ii) is not consistent with the Gaussian integral (4.15). With the stated values alpha = (3/8)R_N^2 and beta = (3/4)R_N^2, the coefficient of k^2 in the actual K2 integral is beta^2/(4alpha^2) = 1, whereas condition (4.17)(ii) requires beta^2/(2alpha^2) = 2. In addition, the exponent written in Eq. (4.20), containing the 9/4 k^2 term, gives gamma = (3/8)R_N^2 and hence gamma - beta^2/(4alpha) = 0, so K1 would not equal exp(-R_N^2 k^2/6) as required by (4.17)(i); it is the exponent in Eq. (4.19), with the 13/4 coefficient, that satisfies that condition. Consequently the removal of the 'spurious' 1/R_N^2 term and the k^2 coefficient quoted in Eq. (4.21) are not correctly derived as written. Because the same K-distribution is used for the scalar and vector matches, this affects both Sec. IV and Sec. VI.
  3. [Sec. VII, Eqs. (7.14), (7.22), (7.23)] The axial-vector spin-orbit matching appears to contain a numerical inconsistency. Equating the (q x k) coefficient in Eq. (7.14), -2 i g'_a/(M'M), with that in Eq. (7.22), i g''_a (4R_N^2/3), gives g''_a = -3g'_a/(2(MR_N)^2), which differs from the value g''_a = -3g_a/(8(MR_N)^2) quoted in Eq. (7.23). If the proportionality sign in Eq. (7.22) absorbs additional coefficients, they should be displayed explicitly; otherwise the axial-vector extra coupling is off by a factor of four. This needs to be reconciled before the axial-vector matching can be considered established.
minor comments (4)
  1. [Abstract and throughout] There are numerous typos and unfinished expressions (for example, 'ratio's' in the abstract, 'the 9NN)' in the abstract, 'LeYouanc' in the introduction, and the garbled condition (4.18b)); the manuscript needs a careful proofreading before resubmission.
  2. [Eq. (3.9b)] The integral J0(q) in Eq. (3.9b) is an integral over S but its exponential contains a q·Q term; this should presumably read q·S, as in Eq. (3.10b).
  3. [Eqs. (6.5) and (6.15)] The phrase 'determine for mu = 0' is confusing because mu is used both as a meson mass and as a Lorentz index; the authors presumably mean the massless-meson limit m = 0 of the form factor.
  4. [Sec. IV C, footnote 1] Footnote 1 compares the K-distribution to exp(-(K-k)^2/epsilon) with epsilon = 8/(3R_N^2), but the sign of the exponent in Eq. (4.20) differs from that in the defining distribution (4.14); the relation should be stated with consistent signs.

Circularity Check

4 steps flagged · score 7.0 of 10

Equality of CQM and NNM vertex coefficients is partly reverse-engineered: extra couplings and the Gaussian K-distribution are fixed by the equality they are meant to establish.

  1. fitted input called prediction [Sec. IV.A, Eq. (4.11)]
    "Compared to Γ NN the quark vertex Γ CQM has an extra 8 k2/(16M′M)-term. This term can be canceled by tuning the g2-coupling. For that purpose we set −g2 9/(16M′M) = g1 8/(16M′M), g2/g1 = −8/9 ≈ −g1."

    The scalar derivative-coupling ratio g2/g1 is not obtained from an independent principle or observable; it is explicitly 'tuned' so that the quark-level k² term cancels the difference between Γ_CQM and Γ_NN. The later claim that the CQM 'reproduces' the nucleon-level scalar vertex therefore rests on a parameter chosen to force that equality. The ratio is an input fitted to the target equality, not a prediction from the model.

  2. fitted input called prediction [Sec. VI.A.1.c, Eqs. (6.5)-(6.6)]
    "This term can be canceled by introducing an extra QQV-interaction, similar to (4.1), ∆H_V^(1)= f′_{1,v}[□(ψ̄γ^μψ)/(2μ²)]V_μ, and determine for μ=0 the coupling from the condition f′_{1,v} 9/(8M′M) = F_{1,v} 8/(16M′M), f′_{1,v}/F_{1,v} = 4/9."

    The new vector interaction is introduced ad hoc and its coupling strength f'_1v is 'determined from the condition' that the quark-level vertex match the nucleon-level vertex. This is reverse-engineering: the equality being claimed is used to define the strength of the extra interaction. The same construction is repeated for f'_2v in Eq. (6.16), so the vector-meson matching is imposed by construction.

2 more flagged steps
  1. fitted input called prediction [Sec. VII.A.4, after Eq. (7.14)]
    "By choosing g′_a = ga, where ga is the axial coupling constant at the quark level, the axial-vertex becomes Γ_{5,CQM} ∼ Γ_{5,NN}."

    The extra axial interaction ΔH (7.13) is introduced phenomenologically, and its strength g'_a is 'chosen' equal to ga precisely to make the spin-orbit part of the quark-level vertex match the nucleon-level one. Equation (7.23) then fixes g''_a = -3g_a/[8(MR_N)^2] so that the orbital-angular-momentum reinterpretation reproduces the already-fitted term. The claimed axial-vector agreement is therefore enforced by these choices rather than derived.

  2. fitted input called prediction [Sec. IV.C, Eqs. (4.17)-(4.19)]
    "and require (i) K1(k2)=exp(−R_N² k²/6), (ii) K2(k2)=(4/R_N²+β²/(2α²)k²)K1(k²), K3,i(k²)=k_i K1(k²). These conditions give ... α=(3/8)R_N², β=(3/4)R_N², and γ=(13/24)R_N²."

    The Gaussian K-distribution, representing momentum correlation, is introduced 'to produce a Γ_CQM without a spurious term', and its parameters α, β, γ are solved from conditions that force K1, K2, K3 to take the exact values needed to cancel the 1/R_N² terms and reproduce the desired Gaussian vertex. Thus the clean vertex (4.21) is an artifact of fitting the distribution to the target, not a consequence of the CQM wave function alone. Note also that the k² coefficient in condition (ii) is 2, whereas the Gaussian integral (4.15) gives coefficient 1; the claimed cancellation is not algebraically checked as written.

full rationale

There is real independent content in the paper: the pseudoscalar vertex (Sec. V) matches without extra couplings, the vector direct term Γ1 (Eqs. 6.9-6.10) follows from the CQM mass relation alone, and the SU(6) spin summation (App. C) is standard. No load-bearing self-citation chain was found; references to the author's ESC work serve as the phenomenological target, not as the derivation. However, the abstract's central claim covers scalar, vector, and axial-vector vertices, and for those the agreement is achieved by parameters fixed by requiring the quark-level vertex to equal the nucleon-level vertex: g2/g1 = -8/9 (4.11), f'_1v/F1v = f'_2v/F2v = 4/9 (6.6, 6.16), g'_a = ga (7.14), g''_a = -3ga/(8(MR_N)^2) (7.23), and α = (3/8)R_N², β = (3/4)R_N², γ = (13/24)R_N² (4.18). These are not independent predictions; they are solved from target conditions such as 'determine the coupling from the condition' and 'require K1=..., K2=...'. Additionally, the K2 condition (4.17)(ii) is internally inconsistent with the actual integral (4.15), so the spurious-term cancellation is not even established as written. The comparison is therefore partially circular: for the channels where the headline claim is strongest, the equality of vertex coefficients is imposed by construction rather than discovered. Score 7 reflects this partial reverse-engineering, not a full identity-by-definition of the entire derivation.

Assumptions & free parameters 5 free parameters · 6 assumptions · 5 invented entities

The central derivation carries a large number of assumptions and tuned ingredients: the constituent quark mass relation, the Gaussian wavefunction, the SU(6) spin-flavor wavefunction, the structure-independence conjecture, the Gaussian momentum correlation, and five extra coupling ratios introduced to cancel mismatches. These are not counted as standard physical constants. The only genuinely external anchors are the nucleon mass and the Gaussian width set near 1 fm.

free parameters (5)
  • Scalar extra-derivative coupling ratio g2/g1 = -8/9
    Introduced in Eq. (4.11) to cancel the excess 8 k^2/(16 M'M) term in the folded scalar vertex. No independent determination.
  • Vector direct extra-coupling ratio f'_1v/F1v = 4/9
    Chosen in Eq. (6.6) to cancel the vector central k^2 mismatch. It is fixed by the condition of matching the nucleon-level vertex.
  • Vector derivative extra-coupling ratio f'_2v/F2v = 4/9
    Chosen in Eq. (6.16) for the derivative vector vertex with the same purpose of canceling the k^2 term.
  • Axial orbital-angular-momentum coupling strengths g'_a/g_a and g''_a/g_a = 1 and -3/(8(M R_N)^2)
    Set in Sec. VII, Eq. (7.14) and Eq. (7.23), to reproduce the axial-vector spin-orbit term. The second depends on the wavefunction width R_N.
  • Nucleon radius parameter R_N = about 1 fm
    Gaussian width of the quark wavefunction, Eq. (3.1). It enters the spurious terms and the axial OAM coupling; set from baryon size, not from the target NN vertex.
assumptions (6)
  • domain assumption Constituent quark mass and non-relativistic spinors: m_Q = M_N/3 and E_i is approximately m_i, so quark Dirac spinors reduce to Pauli form with 1/2m_i factors.
    Section III.A, Eq. (3.2), where the kinetic and potential quark energies are assumed to cancel. This is the load-bearing source of the factor-3 differences.
  • domain assumption Gaussian quark wavefunction for the nucleon: psi proportional to exp[-R_N^2/6 sum(pi-pj)^2].
    Eq. (3.1), taken from Le Yaouanc et al.; all overlap integrals and vertex form factors follow from this.
  • domain assumption SU(6) spin-flavor wavefunction and quark summation rule sum_i sigma_i = sigma_N.
    Used in Secs. IV-VII and Appendix C to convert single-quark spin sums into nucleon spin.
  • ad hoc to paper The conjecture that Pauli-invariant ratios are independent of the nucleon's internal structure.
    Introduction, before Eq. (1); used to declare the 1/R_N^2 terms 'spurious' and remove them. It is a postulate, not derived.
  • ad hoc to paper Momentum correlation between active quark and spectator pair, encoded as a Gaussian K-distribution with alpha=(3/8)R_N^2 and beta=(3/4)R_N^2.
    Sec. IV.C and Appendix B. Needed to eliminate spurious central and spin terms; the correlation is assumed, not derived from QCD.
  • domain assumption Impulse approximation: only one quark per nucleon exchanges the meson; spectator quarks pass through unchanged.
    Overlap integral in App. A, Eq. (A2), with delta functions for spectator quarks; standard quark model approximation.
invented entities (5)
  • Scalar derivative quark-meson coupling g2 square(bar psi psi)/(2 mu^2) sigma
    purpose: Cancels the excess k^2 term in the folded scalar-meson NN vertex.
    Introduced in Sec. IV, Eqs. (4.1)-(4.11); the strength ratio g2/g1=-8/9 is chosen solely to achieve equality with the phenomenological vertex.
  • Vector direct derivative quark coupling f'_1v square(bar psi gamma^mu psi)/(2 mu^2) V_mu
    purpose: Cancels the k^2 mismatch in the vector direct vertex.
    Eqs. (6.5)-(6.6), with f'_1v/F1v=4/9 set by the matching condition.
  • Vector derivative quark coupling f'_2v square(i bar psi partial^mu psi)/(2 mu^2) V_mu
    purpose: Cancels the k^2 mismatch in the derivative vector vertex.
    Eqs. (6.15)-(6.16), with ratio 4/9 chosen to force agreement with the nucleon-level Gamma^0_2 vertex.
  • Axial-vector orbital angular momentum coupling (g'_a and g''_a terms)
    purpose: Reproduces the spin-orbit part of the axial-vector NN vertex via quark orbital angular momentum.
    Sec. VII, Eqs. (7.13)-(7.23); the coupling strengths are set to g_a and -3g_a/(8(MR_N)^2), tuned to match the required q x k term.
  • Gaussian momentum distribution replacing delta^3(K-k) at the QQM vertex
    purpose: Eliminates the spurious 1/R_N^2 terms in the scalar, vector, and axial folded vertices.
    Sec. IV.C, Eqs. (4.14)-(4.19), and App. B; the width parameters alpha, beta, gamma are chosen to cancel unwanted terms.

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Pith. "Pith review of Constituent Quark Model and nucleon-Nucleon Potentials." pith.science (2026). https://pith.science/paper/KGS35PFZ

@misc{pith2026241219858,
  author       = {Pith},
  title        = {Pith review of: Constituent Quark Model and nucleon-Nucleon Potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KGS35PFZ}},
  note         = {Machine review of arXiv:2412.19858}
}
abstract

In these notes, while focusing on the meson-nucleon vertices, we give a derivation of the nucleon-nucleon 9NN) potentials from meson-exchange between quarks. To establish such a relation the quark-quark-meson (QQM) interactions are properly defined. Hitherto, the coefficients in the Pauli-spinor expansion of the meson-nucleon-nucleon (NNM) vertices are equated with those of the QQM-vertices. In these notes we employ the description of the nucleon with Dirac-spinors in the SU(6) semi-relativistic "constituent" quark-model (CQM) as formulated by LeYouanc, et al. It appears that the constituent quark model with $m_q= M_N/3$, is able to produce the same ratio's for the central-, spin-spin-, tensor-, spin-orbit-, and quadratic-spin-orbit Pauli-invariants as in the phenomenological NNM-vertices. In order to achieve this, the scalar-, magnetic-vector, and axial-vector interactions require, besides the standard ones, an extra coupling to the quarks without the introduction of new parameters. in the case of the axial-vector mesons an extra coupling to the quarks is necessary, which is related to the quark orbital angular momentum contribution to the nucleon spin. Furthermore, a momentum correlation between the quark that is coupled to the meson and the remaining quark pair, and a (gaussian) QQM form factor, are necessary to avoid "spurious" terms. From these results we have obtained a formulation of the QQ-interactions which is directly related to the NN extended-soft-core (ESC) interactions. This has been applied to mixed quark-nuclear matter in a study of (heavy) neutron stars.

Figures

Figures reproduced from arXiv: 2412.19858 by the authors.

Figure 1
Figure 1. FIG. 1: Meson-nucleon-nucleon coupling [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Nucleon internal quark momenta with meson-exchange [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Internal quark momenta for meson-exchange [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: External and internal momenta meson-exchange with d [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

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Works this paper leans on

72 extracted references · 70 canonical work pages · cited by 1 Pith paper

  1. [1]

    (6.2) Notice that the 1 /m2 i terms are the same as for scalar-exchange apart from the sign

    Γ 0 1,CQM -vertex: The QQ-meson vertices are [ ¯ui(k′i)γ0ui(ki) ] = √ E′ i +m′ i 2m′ i Ei +mi 2mi ·χ′† i · [ 1 + σ i · k′i E′ i +mi σ i · ki Ei +mi ] ≈ χ′† i [ 1 + k′ i · ki 4m2 i + i 4m2 i σ i · k′ i × ki ] χi = χ′† i [ 1 + Q2 i − k2 16m2 i − i 8m2 i σ i · Qi × k ] χi. (6.2) Notice that the 1 /m2 i terms are the same as for scalar-exchange apart from the...

    work page

  2. [2]

    1: Meson-nucleon-nucleon coupling

    Oi uN1(p1) uN2(p2) ] 2 /B4/CP/B5 /C9/C8/BV/B9/D1/D3 /CS/CT/D0 /B4/CQ/B5 /C6/C6/C5/B9/DA /CT/D6/D8/CT/DC FIG. 1: Meson-nucleon-nucleon coupling. where a complete set independent (t-channel) Lorentz-invariant s can be chosen as O1 = 1 ⊗ 1 O2 =γ5 ⊗γ 5 O3 =γµ ⊗γµ O4 =γ5γµ ⊗γ 5γµ O5 =σµν ⊗σµν O6 =i {γµK µ ⊗ 1 − 1 ⊗γµP µ}, where P = p1 + p′ 1,K = p2 + p′

    work page

  3. [3]

    The Lorentz structure of the NN-amplitude and NNM-vertices given above is general and independent of the internal structure of the nucleon

    We note that Oi = Γ 1,i ⊗ Γ 2,i and that in the meson-exchange contribution to the NN-amplitude the NNM-vertex is of the form ¯u(p′,s ′)Γu(p,s ). The Lorentz structure of the NN-amplitude and NNM-vertices given above is general and independent of the internal structure of the nucleon. Therefore, the QQM-exchange vertices folded with the nucleon quark wave...

    work page

  4. [4]

    (3.4) In (3.4) the γ’s denote the vertex functions

    · ˜ψp1 (k1, k2, k3) ˜ψp2 (q1, q2, q3) · × δ3 (k′2 − k2) δ3 (k′3 − k3) δ3 (q′ 2 − q2) δ3 (q′ 3 − q3) · × γ (k;k′ 1,k 1) γ (k;q′ 1,q 1) k2 +m2 M ·δ3 (k − k′ 1 + k1) δ3 (k + q′ 1 − q1) . (3.4) In (3.4) the γ’s denote the vertex functions. Using the gaussian wave function o f equation (3.1), the overlap integral in Eq. (3.4) can be evaluated in a straightforw...

    work page

  5. [5]

    Γ 1,QQ-vertex: The QQ-meson vertices are [¯ui(k′i)γ iui(ki)] = √ E′ i +m′ i 2m′ i Ei +mi 2mi ·χ′† i · [ σ iσ i · ki Ei +mi + σ i · k′ i σ i E′ i +mi ] χi ≈ χ′† i · [ Qi 2mi + i 2mi (σ i × k) ] χi ⇒ χ′† i · [ q 3mi + i 2mi (σ i × k) ] χi (6.8) Summing over the quarks leads to Γ 1,CQM = ∑ i=1−3 [ ¯ui(k′i)γ iui(¯ki) ] =χ′† i · [ q mi + i 2mi (σ N × k) ] χi (...

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    (7.3) It is clear that this vertex is proportional to that for the OBE-cou pling of the axial-vector meson

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    Γ 5-vertex: The QQ-meson vertices are [¯ui(k′ i)γ iγ5ui(ki)] = √ E′ i +m′ i 2m′ i Ei +mi 2mi ·χ′† i · [ σ i + σ i · k′i σ i σ i · ki (E′ i +mi)(Ei +mi) ] χi ≈χ′† i [ σ i + 1 4m2 i { k′ i(σ i · ki) + ki(σ i · k′ i) − (k′ i · ki)σ i −i(k′ i × ki) }] χi =χ′† i [ σ i + 1 16m2 i { ...

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    Γ 5-vertex(continued A): Next, we impose for the quarks the conse rvation of the axial current. The current is J a µ =ga ¯ψγµγ5ψ + ifa M∂µ( ¯ψγ5ψ), (7.8) 15 and ∂ ·J A = 0 imposes the relation fa = ( m2 A1 2mQM )−1 ga. (7.9) TakingmA1 = √ 2mρ ≈ 2 √ 2mQ the axial current become...

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    Γ 5-vertex(continued B): We note that Γ =∑ 3 i=1 ¯uiγ iγ5ui ⟨¯uN Σ NuN ⟩ for non-relativistic quarks, i.e. it measures the contribution of the quarks to the nucleon spin. In the parton m odel it appeared that a large portion of the nucleon spin has to come from gluonic and qua...

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