Quark–quark hyperfine interactions and their medium modification in the Friedberg–Lee soliton model of nuclear matter and hypernuclei

In the Friedberg–Lee (FL) quark–soliton model coupled self-consistently to scalar and vector meson mean fields, Liang and Shen [Phys. Rev. C 88, 035208 (2013)] obtained parameter-free descriptions of Λ hypernuclei and predicted substantial in-medium swelling of baryon radii, but the model — lacking explicit quark–quark hyperfine interactions — predicts an attractive Σ-nucleus potential of roughly −28 MeV at nuclear saturation density, contradicting the repulsive interior Σ potential inferred from Σ−-atom data and (π−,K+) spectra and the empirical absence of Σ hypernuclei. The open problem is to incorporate one-gluon-exchange-type hyperfine interactions into the FL framework, treat their medium modification self-consistently, and determine whether the resulting model reproduces the observed asymmetry — Λ hypernuclei bound, Σ hypernuclei unbound. A closely related mechanism is already established in the bag-based quark–meson coupling model (Guichon, Thomas, and Tsushima, 2008), but the FL-specific construction and its predictions have never been carried out.

Background

A hypernucleus is an atomic nucleus in which one or more protons or neutrons are replaced by a hyperon — a baryon containing a strange quark. Λ hypernuclei (the Λ has quark content $uds$ and isospin 0) are observed throughout the nuclear chart, bound by a shallow attractive potential of about −30 MeV, whereas no bound Σ hypernuclei are known (the Σ has quark content $uds$ and isospin 1), with the single exception of a quasibound $^{4}_{\Sigma}\mathrm{He}$ state with isospin $I=1/2$ . Analyses of Σ−-atomic level shifts and widths and of inclusive $(\pi^{-},K^{+})$ spectra on medium and heavy targets indicate that the Σ-nucleus potential is repulsive in the nuclear interior (interior repulsion of order 10–40 MeV in distorted-wave and $G$ -matrix analyses — reaction-theory analyses of the spectra and of the in-medium effective two-nucleon interaction, respectively; simple Woods–Saxon fits, i.e. fits with a diffuse-surface radial potential well, use a repulsive real depth of order +100 MeV with a sizable imaginary absorptive part), with at most weak surface attraction; this empirical picture is consolidated in the review by Gal, Hungerford, and Millener (2016). Explaining simultaneously the Λ attraction and the Σ repulsion from first principles is a long-standing challenge.

Relativistic mean-field (RMF) models treat baryons as structureless Dirac particles coupled to classical scalar ( $\sigma$ ) and vector ( $\omega$ , $\rho$ ) meson fields, with the meson–hyperon couplings fixed phenomenologically from suggested potential depths. Quark-level models instead build the medium modification of baryon structure in explicitly. In the quark–meson coupling (QMC) model the baryon is a MIT bag (three quarks confined within a static spherical cavity) that couples directly to the meson mean fields; the scalar field lowers the effective quark masses, changing baryon masses, radii, and spectra in medium. In the Friedberg–Lee (FL) model (Friedberg and Lee, 1977) the baryon is a nontopological soliton: three quarks are confined by a scalar 'composite-gluon' field $\chi$ with a self-interaction potential, so the confinement dynamics is an integral part of the model rather than an external bag boundary. Wen and Shen (2008) coupled the FL soliton to nuclear matter and finite nuclei through the QMC mechanism (quarks coupling to $\sigma$ , $\omega$ , $\rho$ mean fields), and Liang and Shen (2013) extended this to Λ hypernuclei.

In free-space quark models the mass splittings within baryon multiplets, such as $N$ – $\Delta$ and $\Lambda$ – $\Sigma$ , arise from the quark–quark hyperfine interaction — the spin–spin force generated by one-gluon exchange (in the color-symmetric $qq$ pairs relevant for ground-state octet baryons it is repulsive in the $Σ$ and attractive in the $\Lambda$ spin structure). Liang and Shen included only a phenomenological free-space spin-dependent term fixed to the free baryon masses, with no dynamical hyperfine interaction and no in-medium modification of it. Their FL calculation, with no adjustable parameters beyond those fitted to free baryons and nuclear matter saturation, reproduced Λ single-particle energies well with one parameter set (set B) and predicted significant baryon swelling — the root-mean-square nucleon radius growing by about 13% (set B) to 30% (set A) at saturation density, the Λ radius by 9% to 23%, a prediction relevant to the long-debated question of how much nucleon structure swells in nuclei (the EMC effect, i.e. the observed modification of quark momentum distributions in nuclei). But the same model yields a Σ potential of about −28 MeV in saturated nuclear matter, so that Σ hypernuclei would be bound much like Λ hypernuclei, in conflict with the empirical repulsion. The QMC experience shows what is at stake: the early QMC treatment of Λ, Σ, and Ξ hypernuclei by Tsushima, Saito, Haidenbauer, and Thomas (1998) had exactly the same failure (Σ bound like Λ). Guichon, Thomas, and Tsushima (2008) then included the effect of the mean scalar field on the hyperfine interaction self-consistently; the resulting in-medium enhancement of the hyperfine repulsion made all Σ hypernuclei unbound, with a Σ potential repulsive in the interior and weakly attractive at the surface, in qualitative agreement with the Σ-atom analyses, while Λ and Ξ hypernuclei were described as well as before without extra parameters. Whether the same mechanism survives in the FL soliton framework — where confinement is generated by a dynamical field $\chi$ rather than a bag boundary, and where the predicted baryon swelling is large and parameter-set dependent — is the open question. Two further technical limitations acknowledged by Liang and Shen are the local-density approximation (LDA), in which the meson mean fields are frozen at their value at the baryon center and their spatial variation across the baryon volume is neglected, and the use of only the simple relativistic energy–momentum-relation center-of-mass correction instead of a Peierls–Yoccoz projection (a variational projection that removes the spurious collective center-of-mass motion from a localized soliton).

Problem Statement

Within the Friedberg–Lee quark–soliton framework, in which baryons are nontopological solitons of three quarks bound by a scalar confining field and coupled to self-consistent $\sigma$ , $\omega$ , $\rho$ meson mean fields at the quark level, incorporate explicit quark–quark hyperfine interactions of one-gluon-exchange type and treat their medium modification self-consistently. With the extended model calibrated to the free baryon spectrum (including the $\Lambda$ – $\Sigma$ splitting) and to nuclear matter saturation, as in the source calculation, determine:

(i) how the hyperfine interaction and the resulting $\Lambda$ – $\Sigma$ mass splitting are modified in nuclear matter as a function of density;

(ii) whether the resulting Σ-nucleus potential becomes repulsive in the nuclear interior — at the scale of roughly 10–40 MeV indicated by Σ−-atom and $(\pi^{-},K^{+})$ analyses — so that Σ hypernuclei are unbound or at most quasibound, while Λ hypernuclei remain bound at the observed level.

The common objective of both parts is a quantitatively controlled Friedberg–Lee-model description of medium-modified baryon structure that reproduces the observed asymmetry between Λ and Σ hypernuclei without adjustable hyperon–meson couplings. The local-density approximation and the relativistic energy–momentum-relation center-of-mass correction of the source calculation are retained; lifting them is not required for the description to count as quantitatively controlled.

References

J. S. Liang and H. Shen, 'Medium modifications of baryon properties in nuclear matter and hypernuclei', Phys. Rev. C 88, 035208 (2013). DOI: 10.1103/PhysRevC.88.035208, https://doi.org/10.1103/PhysRevC.88.035208 (preprint arXiv:1304.4713, https://arxiv.org/abs/1304.4713).
R. Friedberg and T. D. Lee, 'Fermion-field nontopological solitons', Phys. Rev. D 15, 1694 (1977). DOI: 10.1103/PhysRevD.15.1694, https://doi.org/10.1103/PhysRevD.15.1694.
R. Friedberg and T. D. Lee, 'Fermion-field nontopological solitons. II. Models for hadrons', Phys. Rev. D 16, 1096 (1977). DOI: 10.1103/PhysRevD.16.1096, https://doi.org/10.1103/PhysRevD.16.1096.
P. A. M. Guichon, A. W. Thomas, and K. Tsushima, 'Binding of hypernuclei in the latest quark-meson coupling model', Nucl. Phys. A 814, 66 (2008). DOI: 10.1016/j.nuclphysa.2008.10.001, https://doi.org/10.1016/j.nuclphysa.2008.10.001 (preprint arXiv:0712.1925, https://arxiv.org/abs/0712.1925).
K. Tsushima, K. Saito, J. Haidenbauer, and A. W. Thomas, 'The quark-meson coupling model for Λ, Σ and Ξ hypernuclei', Nucl. Phys. A 630, 691 (1998). DOI: 10.1016/S0375-9474(98)00806-9, https://doi.org/10.1016/S0375-9474(98)00806-9 (preprint arXiv:nucl-th/9707022, https://arxiv.org/abs/nucl-th/9707022).
A. Gal, E. V. Hungerford, and D. J. Millener, 'Strangeness in nuclear physics', Rev. Mod. Phys. 88, 035004 (2016). DOI: 10.1103/RevModPhys.88.035004, https://doi.org/10.1103/RevModPhys.88.035004 (preprint arXiv:1605.00557, https://arxiv.org/abs/1605.00557).
W. Wen and H. Shen, 'Modification of nucleon properties in nuclear matter and finite nuclei', Phys. Rev. C 77, 065204 (2008). DOI: 10.1103/PhysRevC.77.065204, https://doi.org/10.1103/PhysRevC.77.065204.
J. N. Hu, A. Li, H. Shen, and H. Toki, 'Quark mean-field model for single and double Λ and Ξ hypernuclei', Prog. Theor. Exp. Phys. 2014, 13D02 (2014). DOI: 10.1093/ptep/ptt119, https://doi.org/10.1093/ptep/ptt119 (preprint arXiv:1310.3602, https://arxiv.org/abs/1310.3602).
K. Azizi and N. Er, 'Properties of Σ_Q^, Ξ_Q^ and Ω_Q^* heavy baryons in cold nuclear matter', Nucl. Phys. A 970, 422 (2018). DOI: 10.1016/j.nuclphysa.2018.01.006, https://doi.org/10.1016/j.nuclphysa.2018.01.006 (preprint arXiv:1801.02168, https://arxiv.org/abs/1801.02168).
N. Er and K. Azizi, 'Fate of the doubly heavy spin-3/2 baryons in a dense medium', Phys. Rev. D 99, 074012 (2019). DOI: 10.1103/PhysRevD.99.074012, https://doi.org/10.1103/PhysRevD.99.074012 (preprint arXiv:1901.07399, https://arxiv.org/abs/1901.07399).
H. Nemura, S. Aoki, T. Doi, S. Gongyo, T. Hatsuda, Y. Ikeda, T. Inoue, T. Iritani, N. Ishii, T. Miyamoto, K. Murano, and K. Sasaki (HAL QCD Collaboration), 'Lambda-Nucleon and Sigma-Nucleon interactions from lattice QCD with physical masses', PoS LATTICE2016, 101 (2017). DOI: 10.22323/1.256.0101, https://doi.org/10.22323/1.256.0101 (preprint arXiv:1702.00734, https://arxiv.org/abs/1702.00734).