Extended-soft-core Baryon-Baryon Model ESC08 I. Nucleon-Nucleon Scattering

Abstract: The Nijmegen extended-sft-core ESC08c model for the baryon-baryon (BB) interactions of the SU(3) flavor-octet of baryons ($N, Λ, Σ$, and $Ξ$) is presented. In this first of a series of papers, the NN results are reported in detail. In the spirit of the Yukawa-approach to the nuclear force problem, the interactions are studied from the meson-exchange picture viewpoint, using generalized soft-core Yukawa-functions. These interactions are supplemented with (i) multiple-gluon-exchange, and (ii) structural effects due to the quark-core of the baryons. The extended-soft-core (ESC) meson-exchange interactions consist of local- and non-local-potentials due to ((i) One-boson-exchanges (OBE, which are the members of nonets of pseudoscalar , vector, scalar, and axial-vector mesons, (ii) diffractive (i.e. multiple-gluon) exchanges, (iii) two pseudoscalar exchange (PS-PS), and (iv) meson-pair-exchange (MPE). The OBE- and MPE-vertices are regulated by gaussian form factors producing potentials with a soft behavior near the origin. The assignment of the cut-off masses for the BBM-vertices is dependent on the SU(3)-classification of the exchanged mesons for OBE, and a similar scheme for MPE. The simultaneous fit of the ESC-models to the NN- and YN-scattering data achieved excellent results for the NN, YN, and favorable properties for the $ΛΛ$ and $ΞN$ systems. In the case of ESC08c, the version of this paper, the results are: (i) For the selected 4313 pp and np scattering data ($ 0 \leq T_{lab} \leq 350$ MeV), the model achieved $χ^2/N_{data} = 1.08$. (ii) The deuteron binding energy and all NN low energy parameters are fitted very nicely. (iii) The YN-data are described also very well with $χ^/N_{data} = 1.09$. (iv) The model predicts a bound $ΞN(^3S_1,I=1)$ states with binding energy 1.56 MeV.

• The paper presents a unified meson-exchange framework with integrated quark-core and multi-gluon effects to accurately describe NN scattering.
• It employs simultaneous χ² fitting across NN, YN, and YY data, reproducing experimental scattering lengths, effective ranges, and deuteron binding energy.
• The ESC08 model’s adherence to SU(3) symmetry and QCD-inspired constraints ensures predictive power for hypernuclear matter and neutron star equations of state.

Extended-Soft-Core Baryon-Baryon Model ESC08: Nucleon-Nucleon Scattering

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Introduction and Motivation

The paper presents the ESC08c version of the Nijmegen extended-soft-core (ESC) model for baryon-baryon (BB) interactions, addressing nucleon-nucleon (NN) scattering and, in future parts, interactions involving additional strangeness ($S = -1,-2,-3,-4$). The theoretical framework is grounded in a meson-exchange picture augmented by explicit inclusion of multi-gluon processes (pomeron and odderon exchange) and quark-core effects (phenomenological Pauli-blocking contributions).

A key motivation is to provide a robust, physically constrained, and unified description for NN, hyperon-nucleon (YN), and hyperon-hyperon (YY) channels—using a single set of meson-quark parameters, linked by broken $SU(3)$ flavor symmetry, and fitted to all available low-energy scattering and hypernuclear data. This approach enables predictive modeling of poorly constrained channels (especially those with $S=-2$ and below), relevant for hyperonic matter, neutron-star EoS, and finite hypernuclei.

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Model Construction: Dynamical Ingredients

The ESC08 model describes the baryon-baryon interaction as a superposition of:

1. One-Boson Exchange (OBE):
   • Inclusion of nonets for pseudoscalar, vector, scalar, and both kinds of axial-vector mesons.
   • Gaussian form factors with flavor-dependent cut-offs ensure physically motivated suppression at short distances.
   • Novelty: a "zero" is introduced in the form factor for scalar and axial-vector mesons reflecting their P-wave $Q\bar{Q}$ structure (as required by the $^3P_0$ quark-pair-creation model).
2. Two-Meson Exchange (TME):
   • Uncorrelated two-pseudoscalar exchanges computed in a fully $SU(3)_f$-invariant way (extended to all relevant combinations for BB systems).
3. Meson-Pair Exchange (MPE):
   • Low-energy effective interactions involving two-meson pairs; MPE couplings inferred via heavy-meson and baryon-resonance saturation and scaled according to $SU(3)$.
4. Diffractive/Multi-Gluon Exchange:
   • Pomeron ($J^{PC}=0^{++}$, even-gluon) and odderon ($J^{PC}=1^{--}$, odd-gluon, new to ESC08) contributions are modeled by gaussian potentials.
   • The odderon term is essential for central and spin-dependent repulsion at short distances.
5. Quark-Core Effects (QCE):
   • Extra short-range repulsion incorporated as gaussian contributions, weighting the contribution channelwise according to the presence of forbidden six-quark states (SU(6)-32 symmetry), mimicking Pauli-blocked configurations realized in quark-basis representations.
   • This parametric addition resolves deficiencies in describing strong short-range repulsion in $\Sigma N(^3S_1)$, $\Sigma N(^1S_0)$, and $\Xi N$ channels.

All interactions are systematically regulated at short distance and cast into (momentum or configuration) space via convolution with the chosen gaussian form factors, yielding well-defined, numerically stable potentials. The OBE sector models both local and nonlocal (momentum-dependent) contributions, ensuring that high-momentum components are physically suppressed.

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Coupling Parameters, Symmetry Constraints, and QPC Consistency

A central theme is the close connection between the fitted baryon-baryon-meson (BBM) coupling constants and predictions from the $^3P_0$ ($+$ $^3S_1$ admixture) quark-antiquark pair creation model (QPC). For OBE couplings and pair vertices, the $F/(F+D)$ (octet/singlet ratio) parameters are chosen so as to deviate minimally from QPC values. SU(3) relations are imposed to relate NN, YN, and YY couplings; the true fit is thus posed as a highly nontrivial, constrained optimization over a reduced set of physically interpretable parameters.

For example, in ESC08c, all OBE couplings are constrained within the QPC predictions: the model’s fitted values for pseudoscalar, vector, scalar, and axial-vector couplings are in quantitative agreement with $^3P_0+^3S_1$ QPC predictions for $r_M\sim0.7-1$ fm, with minor SU(6) symmetry breaking further improving the fit. Nonstandard OBE terms—such as the inclusion of derivative axial-vector and a zero in P-wave $Q\bar{Q}$ form factors—enable good phenomenology with couplings much closer to hadron-physics priors.

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Simultaneous Fitting Protocol and Numerical Achievements

A critical aspect of ESC08 is the simultaneous $\chi^2$ optimization to NN, YN, and available YY (hypernuclear) data, with a single parameter set. The NN data set comprises 4313 $pp$ and $np$ phase-shift data up to $T_{lab}=350$ MeV, using the Nijmegen partial-wave analysis as benchmark. Fitting also includes the deuteron binding energy, low-energy scattering lengths, and effective ranges for all physical channels. The following performance metrics are achieved:

• $\chi^2/N_{data} = 1.08$ for NN data, competitive with purely phenomenological models (NijmI/II/Reid93—with many more free parameters).
• Deuteron binding energy ($E_B=2.224593$ MeV) and S-wave scattering lengths/effective ranges are reproduced within experimental uncertainties.
• Charge independence/symmetry breaking in $^1S_0$ is included at the level of adjusted $\rho$ couplings.
• $YN$ and hypernuclear well depths $U_{Y}$ ($Y=\Lambda,\Sigma,\Xi$) are simultaneously reproduced.
• A bound state is predicted in the $\Xi N(^3S_1, I=1)$ channel, with binding energy $1.56$ MeV.

The successful fit is maintained under stringent physical constraints, including the absence of spurious $S=-1$ ($YN$) bound states and realistic G-matrix results for hypernuclear well depths.

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Theoretical Implications and Extensions

ESC08 robustly demonstrates the viability of an $SU(3)$-connected, physically motivated meson-exchange approach for quantitative baryon-baryon interactions over a broad range of strangeness channels. Key theoretical implications include:

• The direct confirmation that meson-exchange picture, including essential QCD-inspired elements (multi-gluon exchange, quark-core effects), can produce predictive, phenomenologically viable, and physically meaningful NN and YN/YY interactions with a single parameterization.
• The structure of the potentials, including regulated short-distance dynamics, ensures their direct usability in many-body and finite-nucleus calculations.
• The presence of bound $\Xi N$ states and strong tensor forces suggests precise experimental signals for future strangeness nuclear physics programs.
• The multi-gluon exchange sector (odderon and pomeron) and QCE are critical for reproducing both two-body scattering and bulk matter properties (saturation, EoS).

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Application: Nuclear Saturation, EoS, and Three-Body Forces

The analysis is extended to the description of nuclear saturation and the equation of state (EoS) for both symmetric and neutron matter. Free-space two-body potentials from ESC08 are supplemented by density-dependent universal three-body repulsion (triple- and quartic-pomeron exchange, MPP) and phenomenological three-nucleon attraction (TNA):

• The inclusion of universal MPP (flavor-blind, as required by multi-gluon coupling) ensures stiffening of the EoS at high densities, essential for reproduction of empirical saturation density and energy, as well as the observed neutron-star masses ($>2M_\odot$).
• Parameter sets for $g_P^{(3)}$ and $g_P^{(4)}$ are extracted by folding into G-matrix and finite-nucleus calculations, modulating nuclear incompressibility ($K=260$–$310$ MeV) and symmetry energy.
• Hyperon mixing and three-body repulsion are shown to compete, dictating the high-density EoS and neutron star core composition.

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Future Directions

Given the transparent linkage between coupling constants and $Q\bar{Q}$-model predictions, and the simultaneous fit to scattering and hypernuclear observables, the model sets a strong basis for:

• Extension to S=-3,-4 transitions (fully predictive, with no fitted parameters due to $SU(3)$ extrapolation).
• Systematic many-body calculations for hyperonic matter, neutron stars, and finite hypernuclei.
• Exploration of the interplay of universal three-baryon forces and hyperon mixing.
• Investigation of the detailed role of specific meson-exchange components in spin-orbit and tensor observables, and in the evolution of shell structure in the presence of strangeness.

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Conclusion

The ESC08 model, as detailed in this work, constitutes a quantitatively precise, theoretically grounded framework for baryon-baryon interactions, incorporating both QCD- and quark-structure-motivated extensions to the meson-exchange program. Its highly constrained and physically transparent parameterization achieves an excellent description of all low-energy NN and YN phenomenology, provides robust predictions for strangeness channels, and enables consistent nuclear matter and neutron-star applications. The close correspondence of the fitted couplings to quark model predictions, the successful implementation of universal repulsion mechanisms, and the proper inclusion of quark-core and multi-gluon exchange effects, collectively substantiate ESC08 as a leading tool for both phenomenological and theoretical studies in hadronic and nuclear physics.

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