Dynamical QCD origin of the constituent quark's Dirac-like electroweak couplings: the $g_A=+1$ selection and the vanishing anomalous magnetic moment (Weinberg, 1990)

In the constituent quark model, the light $u,d$ quarks act as heavy pointlike Dirac particles with axial coupling $g_A=1$ and zero anomalous magnetic moment, even though QCD dressing by gluons and quark-antiquark pairs completely changes their masses. Weinberg (Phys. Rev. Lett. 65, 1181 (1990)) showed that, if Adler-Weisberger and photoproduction sum rules can be saturated by narrow one-particle states in the colored one-quark sector and if there are no other color-triplet particles, the chiral algebra forces $g_A=\pm1$ (and, allowing excited quark states together with large-$N_c$ planar counting and high-energy forward elasticity of pion-quark scattering, restricts $g_A$ to $+1,0,-1$), with $g_A=+1$ implying vanishing anomalous magnetic moments. The paper itself left open why $g_A=+1$ is realized rather than $0$ or $-1$, whether the saturation, elasticity, and large-$N_c$ assumptions hold at $N_c=3$, what excited color-triplet states would do, and why the argument does not extend to nucleons. Later literature computed model-level corrections ($g_A^q\approx0.75$-0.91), identified mechanisms producing $O(N_c^0)$ deviations ($\pi$-$a_1$ mixing, vector mesons, subtraction constants in the underlying dispersion relation), and supported the $O(1/N_c)$ suppression of the anomalous magnetic moment by chirality arguments, but no dynamical QCD derivation of the selection, the exact value, or the assumption status at $N_c=3$ exists, and the baryon extension remains unsettled.

Background

In QCD with two light quark flavors, the $u$ and $d$ quarks enter the Lagrangian nearly massless, yet low-energy hadron phenomenology is dominated by the constituent quark model, in which each quark behaves as a particle of mass $\sim300$ MeV. This mass is generated dynamically: spontaneous breakdown of the chiral symmetry $\mathrm{SU}(2)\otimes\mathrm{SU}(2)$ (the symmetry connecting left- and right-handed quark fields, broken spontaneously by the QCD vacuum) dresses the quark with a cloud of gluons and $q\bar q$ pairs whose Goldstone bosons (the massless excitations that any spontaneously broken symmetry necessarily generates) are the pions. The chiral quark model of Manohar and Georgi embeds constituent quarks and pions in a single effective theory and simply postulates the quark's axial coupling. The puzzle addressed by Weinberg (1990) is that the same dressing which radically alters the quark mass apparently leaves its other electroweak properties untouched: constituent-quark (and bag-model, with quarks confined in a spherical cavity) calculations reproduce hadronic electroweak observables by treating the constituent quark as a bare Dirac particle, i.e. with axial coupling $g_A=1$ and no anomalous magnetic moment. Here $g_A$ is the coefficient of the weak axial current's matrix element between the particle's spin states, equal to 1 for a free Dirac fermion, and the anomalous magnetic moment is the deviation of the magnetic moment from the Dirac value $2e_q/m_q$.

Weinberg's strategy was to take over into the constituent quark model the familiar apparatus of soft-pion theorems and sum rules. The Adler-Weisberger sum rule (originally relating $1/g_A^2$ of a target to an integral over pion-target total cross section differences) and the Drell-Hearn-type photoproduction dispersion relations (which relate a particle's anomalous magnetic moment to an integral over the difference of the two spin orientations' photon-absorption cross sections), when saturated by narrow one-particle intermediate states, reduce to Lie-algebraic commutators $[X_a,X_b]=i\epsilon_{abc}T_c$, $[\kappa,X_3]=0$, $[\kappa,\kappa^\dagger]=0$, where $X_a$ are reduced pion-emission amplitudes, $T_a$ the isospin matrices, and $\kappa$ the momentum-independent matrix of collinear photon transition amplitudes. If there are no color-triplet particles besides the quark itself, the only representations of the chiral algebra containing a single isospin-$1/2$ multiplet are $(1/2,0)$ and $(0,1/2)$ (the two irreducible representations of $\mathrm{SU}(2)\otimes\mathrm{SU}(2)$ with definite left- and right-handed chirality), which forces $X_a=\pm T_a$ and hence $g_A=\pm1$; for $g_A=+1$ the photoproduction algebra then forces the diagonal element of $\kappa$ to vanish, i.e. vanishing anomalous magnetic moments for the $u$ and $d$ constituent quarks.

Allowing excited quark states (all of isospin $1/2$, hence mixtures of $(1/2,0)$ and $(0,1/2)$), the absence of isospin-2 Regge trajectories (families of hadrons whose spin grows linearly with the square of their mass) yields superconvergence relations — constraints from the high-energy falloff of forward scattering amplitudes — which force the quark mass-squared matrix to decompose into a block-diagonal scalar part $m_S^2$ and an off-diagonal vector part $m_V^2$ in the $(1/2,0)\oplus(0,1/2)$ basis. In the large-$N_c$ limit ($N_c$ = number of colors, $N_c\to\infty$), 't Hooft's planar-diagram topology (only planar Feynman diagrams, drawable without line crossings, survive at leading order) implies that only single color-triplet particles contribute at leading order as intermediate states in pion-quark scattering. If one further assumes that forward pion-quark scattering becomes purely elastic at high energy (an approximation that works well for ordinary hadrons), the matrices $m_S^2$ and $m_V^2$ commute, and a quark mass eigenstate is either a pure $(1/2,0)$ state with $g_A=+1$, a pure $(0,1/2)$ state with $g_A=-1$, or a paired mixture with $g_A=0$ — the only three possibilities. Weinberg stated explicitly that he did not know why the quark should have $g_A=+1$ rather than $0$ or $-1$.

For baryons the argument breaks down: even at large $N_c$, intermediate states in pion-nucleon scattering can be built from $N_c$ quarks and carry isospin $3/2$ (already for $N_c=3$), so the nucleon and its excitations involve a larger set of chiral representations, $(3/2,0)$, $(0,3/2)$, $(1,1/2)$, $(1/2,1)$, $(1/2,0)$, $(0,1/2)$, and the simple commutator algebra yields no model-independent conclusion about the nucleon's empirical axial coupling $g_A^N\approx1.27$ or its magnetic structure.

A structural caveat shapes the whole question: the constituent quark is a colored, confined object, so its axial coupling and anomalous magnetic moment are not gauge-invariant observables; they are effective parameters defined only within a specified low-energy effective theory or model embedding, and they are constrained only indirectly through quantities such as the nucleon axial coupling, heavy-meson decays, and baryon magnetic moments.

Problem Statement

The problem is to replace the assumptions of Weinberg's explanation by QCD dynamics, for QCD with $N_c=3$ colors and two light quark flavors. Its single objective is a QCD-based derivation of the constituent quark's electroweak couplings — the axial coupling and the anomalous magnetic moment — that does not assume the Dirac values, with all of the following jointly discharged:

(1) Selection and sign: give a dynamical account of why the constituent quark realizes $g_A=+1$ among the three values $\{+1,0,-1\}$ allowed by the chiral algebra under the stated assumptions, rather than leaving the selection unexplained.

(2) Assumption status and size of corrections: establish, refute, or control each load-bearing assumption — (a) saturation of the Adler-Weisberger and Drell-Hearn-type sum rules by narrow one-particle states in the colored one-quark sector, including the question of subtractions in the underlying dispersion relations; (b) the role of excited color-triplet states (quark-plus-gluon excitations) and possible colored bound states as intermediate states; (c) the high-energy forward elasticity of pion-quark scattering; (d) the adequacy of large-$N_c$ counting at $N_c=3$ — and thereby determine, with controlled errors, the actual value of the constituent quark's axial coupling and anomalous magnetic moment, i.e. the size of the deviations from 1 and from 0.

(3) Extension to baryons: determine whether comparably rigorous arguments can fix the nucleon's axial coupling ($g_A^N\approx1.27$) and its magnetic structure, given that baryons involve chiral representations beyond $(1/2,0)$ and $(0,1/2)$.

Because the colored, confined constituent quark's couplings are effective parameters defined only within a specified scheme, the derivation is meaningful only once that scheme is fixed; its results are then to be related to the indirect phenomenological anchors (nucleon axial coupling, baryon magnetic moments, heavy-meson decay couplings).

References

1. S. Weinberg, 'Why do quarks behave like bare Dirac particles?', Phys. Rev. Lett. 65, 1181-1183 (1990). DOI: 10.1103/PhysRevLett.65.1181, https://doi.org/10.1103/PhysRevLett.65.1181
2. S. Weinberg, 'Mended symmetries', Phys. Rev. Lett. 65, 1177-1180 (1990). DOI: 10.1103/PhysRevLett.65.1177, https://doi.org/10.1103/PhysRevLett.65.1177
3. A. Manohar, H. Georgi, 'Chiral quarks and the non-relativistic quark model', Nucl. Phys. B 234, 189-212 (1984). DOI: 10.1016/0550-3213(84)90231-1, https://doi.org/10.1016/0550-3213(84)90231-1
4. G. 't Hooft, 'A planar diagram theory for strong interactions', Nucl. Phys. B 72, 461-473 (1974). DOI: 10.1016/0550-3213(74)90154-0, https://doi.org/10.1016/0550-3213(74)90154-0
5. S. Peris, 'gA in the constituent quark model', Phys. Lett. B 268, 415-418 (1991). DOI: 10.1016/0370-2693(91)91599-Q, https://doi.org/10.1016/0370-2693(91)91599-Q
6. S. Weinberg, 'Axial vector coupling of the quark', Phys. Rev. Lett. 67, 3473-3474 (1991). DOI: 10.1103/PhysRevLett.67.3473, https://doi.org/10.1103/PhysRevLett.67.3473
7. D. A. Dicus, Dj. Minic, U. van Kolck, R. Vega, 'The axial vector coupling and magnetic moment of the quark', Phys. Lett. B 284, 384-389 (1992). DOI: 10.1016/0370-2693(92)90449-E, https://doi.org/10.1016/0370-2693(92)90449-E
8. S. Peris, 'Adler-Weisberger sum rule and the axial-vector coupling of a constituent quark', Phys. Rev. D 46, 1202-1204 (1992). DOI: 10.1103/PhysRevD.46.1202, https://doi.org/10.1103/PhysRevD.46.1202
9. W. Broniowski, M. Lutz, A. Steiner, 'Nc-counting rules and the axial vector coupling constant of the constituent quark', Phys. Rev. Lett. 71, 1787-1790 (1993). DOI: 10.1103/PhysRevLett.71.1787, https://doi.org/10.1103/PhysRevLett.71.1787 (arXiv: hep-ph/9304292)
10. S. Peris, E. de Rafael, 'Constituent quark couplings and QCD in the large Nc limit', Phys. Lett. B 309, 389-393 (1993). DOI: 10.1016/0370-2693(93)90950-M, https://doi.org/10.1016/0370-2693(93)90950-M (arXiv: hep-ph/9304262)
11. X. Li, Y. Liao, 'The electroweak couplings of constituent quarks in the nonlocal chiral quark model', Phys. Lett. B 318, 537-543 (1993). DOI: 10.1016/0370-2693(93)91552-X, https://doi.org/10.1016/0370-2693(93)91552-X
12. P. Sieber, M. Praszalowicz, K. Goeke, 'Dependence of baryonic observables on the quark-axial-vector coupling in the chiral quark model', Nucl. Phys. A 569, 629-644 (1994). DOI: 10.1016/0375-9474(94)90377-8, https://doi.org/10.1016/0375-9474(94)90377-8
13. J. Ellis, Y. Frishman, A. Hanany, M. Karliner, 'Quark solitons as constituents of hadrons', Nucl. Phys. B 382, 189-212 (1992). DOI: 10.1016/0550-3213(92)90183-C, https://doi.org/10.1016/0550-3213(92)90183-C
14. P. H. Damgaard, R. Sollacher, 'Qualitons from QCD', Phys. Lett. B 322, 131-140 (1994). DOI: 10.1016/0370-2693(94)90501-0, https://doi.org/10.1016/0370-2693(94)90501-0
15. G. Gomelski, M. Karliner, S. B. Selipsky, 'Static properties of quarl solitons', Phys. Lett. B 323, 182-187 (1994). DOI: 10.1016/0370-2693(94)90289-5, https://doi.org/10.1016/0370-2693(94)90289-5
16. K.-T. Chao, H.-Q. Zheng, 'A note on the constituent quark resonances', Phys. Lett. B 343, 377-380 (1995). DOI: 10.1016/0370-2693(94)01425-C, https://doi.org/10.1016/0370-2693(94)01425-C
17. N.-G. Chen, K.-T. Chao, 'The quark axial vector coupling and heavy meson decays', Phys. Lett. B 345, 67-73 (1995). DOI: 10.1016/0370-2693(94)01578-Z, https://doi.org/10.1016/0370-2693(94)01578-Z (arXiv: hep-ph/9502410)
18. H. Ito, 'Flavor SU(3) symmetry in the anomalous magnetic moments of constituent quarks', Phys. Lett. B 353, 13-19 (1995). DOI: 10.1016/0370-2693(95)00443-O, https://doi.org/10.1016/0370-2693(95)00443-O
19. A. Hosaka, D. Jido, M. Oka, 'Determination of the axial coupling constant gA in the linear representations of chiral symmetry', arXiv:hep-ph/0112355 (2001). DOI: 10.48550/arXiv.hep-ph/0112355, https://arxiv.org/abs/hep-ph/0112355 (also published as a book chapter, DOI: 10.1142/9789812777218_0063, https://doi.org/10.1142/9789812777218_0063)
20. S. R. Beane, M. J. Savage, 'A conjecture about hadrons', Phys. Lett. B 556, 142-152 (2003). DOI: 10.1016/S0370-2693(03)00127-8, https://doi.org/10.1016/S0370-2693(03)00127-8 (arXiv: hep-ph/0212106)
21. W. Lucha, D. Melikhov, S. Simula, 'Constituent quarks, chiral symmetry, and chiral point of the constituent quark model', Phys. Rev. D 74, 054004 (2006). DOI: 10.1103/PhysRevD.74.054004, https://doi.org/10.1103/PhysRevD.74.054004 (arXiv: hep-ph/0606281)
22. S. S. Afonin, 'Axial coupling from matching a constituent quark model to QCD', Phys. Rev. C 77, 058201 (2008). DOI: 10.1103/PhysRevC.77.058201, https://doi.org/10.1103/PhysRevC.77.058201 (arXiv: 0805.2833)
23. S. Weinberg, 'Pions in Large-N Quantum Chromodynamics', Phys. Rev. Lett. 105, 261601 (2010). DOI: 10.1103/PhysRevLett.105.261601, https://doi.org/10.1103/PhysRevLett.105.261601 (arXiv: 1009.1537)
24. Y. Hidaka, T. Kojo, L. McLerran, R. D. Pisarski, 'The dichotomous nucleon: Some radical conjectures for the large Nc limit', Nucl. Phys. A 852, 155-174 (2011). DOI: 10.1016/j.nuclphysa.2011.01.008, https://doi.org/10.1016/j.nuclphysa.2011.01.008 (arXiv: 1004.2261)
25. E. de Rafael, 'The Constituent Chiral Quark Model revisited', Phys. Lett. B 703, 60-65 (2011). DOI: 10.1016/j.physletb.2011.07.046, https://doi.org/10.1016/j.physletb.2011.07.046 (arXiv: 1107.0226)
26. T. Kojo, 'Can the nucleon axial charge be O(Nc^0)?', Nucl. Phys. A 899, 76-106 (2013). DOI: 10.1016/j.nuclphysa.2012.12.116, https://doi.org/10.1016/j.nuclphysa.2012.12.116 (arXiv: 1208.5661)