Lambda Single-Particle Potential

• Lambda Single-Particle Potential is defined as the mean-field interaction experienced by a Lambda hyperon in nuclear matter or finite nuclei, derived from microscopic and phenomenological models.
• It exhibits strong density and momentum dependence, with empirical depths near -30 MeV at saturation and significant repulsion at higher densities, influencing hypernuclear structure and neutron-star equations of state.
• Ab initio methods using chiral EFT and lattice QCD incorporate two- and three-body forces to reconcile hypernuclear data, playing a key role in addressing the hyperon puzzle in astrophysics.

The Lambda ($\Lambda$) single-particle potential, $U_\Lambda$, characterizes the mean-field interaction experienced by a $\Lambda$ hyperon moving in nuclear matter or a finite nucleus. It encodes key information about $\Lambda$–nucleon ($\Lambda N$) and $\Lambda$–nucleon–nucleon ($\Lambda NN$) interactions, exhibits strong density and momentum dependence, and plays a central role in hypernuclear structure, heavy-ion dynamics, and neutron-star equations of state.

1. Formal Definition and Theoretical Frameworks

The $\Lambda$ single-particle potential is typically defined within Brueckner–Hartree–Fock (BHF/G-matrix) or mean-field theory. In infinite, isospin-symmetric nuclear matter at baryon density $\rho$, the standard microscopic expression is: $$U_\Lambda(k_\Lambda; \rho) = \sum_{|\mathbf{k}_N|\leq k_F} \langle \Lambda(\mathbf{k}_\Lambda)N(\mathbf{k}_N) | G_{\Lambda N}(\omega) | \Lambda(\mathbf{k}_\Lambda)N(\mathbf{k}_N) \rangle_A$$ where $k_F$ is the nuclear Fermi momentum, $G_{\Lambda N}(\omega)$ is the in-medium $\Lambda N$ reaction matrix, and $A$ denotes antisymmetrization in the nucleon leg (Jinno et al., 4 Feb 2026). The starting energy $\omega$ includes the self-consistent $\Lambda$ and nucleon mean fields.

Within chiral effective field theory (EFT), modern hyperon interactions include two-body ($\Lambda N$) and density-dependent three-body ($\Lambda NN$) forces, the latter typically entering at next-to-next-to-leading order (N$^2$LO) or beyond (Jinno et al., 27 Aug 2025, Jinno et al., 16 Jan 2025, Haidenbauer et al., 2016). The effective potential is separated into two- and three-body driven components: $$U_\Lambda(p, \rho) = U_\Lambda^{(2)}(p, \rho) + U_\Lambda^{(3)}(p, \rho)$$ with $U_\Lambda^{(2)}$ directly from the bare $\Lambda N$ potential and $U_\Lambda^{(3)}$ from normal-ordering the $\Lambda NN$ three-body force into a density-dependent two-body term (Jinno et al., 27 Aug 2025).

In finite nuclei, $U_\Lambda$ is extracted from the real part of the $\Lambda$ self-energy in perturbative many-body or mean-field models, frequently parametrized as Woods–Saxon or folded-Gaussian potentials (Vidana, 2016, Friedman et al., 2023).

2. Density and Momentum Dependence

The canonical observable is the depth of $U_\Lambda$ at zero momentum and saturation density ($\rho_0 \approx 0.16-0.17\,\mathrm{fm}^{-3}$), which is empirically $-30 \pm 3$ MeV from hypernuclear separation energies (Friedman et al., 2023, Inoue et al., 2016, Jinno et al., 16 Jan 2025, Jinno et al., 27 Aug 2025). Ab initio approaches using chiral EFT YN interactions consistently reproduce this, e.g., $U_\Lambda(\rho_0,0) = -33$ MeV (HAL QCD-lattice+BHF) (Inoue et al., 2016) and $U_\Lambda(\rho_0,0) = -27.3 \pm 0.6$ MeV (global optical fit) (Friedman et al., 2023).

Higher-density behavior is nontrivial. Modern results employing chiral $\Lambda N$ and $\Lambda NN$ interactions show that $U_\Lambda$ becomes progressively less attractive with increasing $\rho$, crossing zero at $\rho \sim 2\,\rho_0$ and becoming strongly repulsive at $3\,\rho_0$ (see Table below) (Jinno et al., 16 Jan 2025, Jinno et al., 27 Aug 2025, Haidenbauer et al., 2016, Kohno, 2018, Friedman et al., 2023).

$\rho/\rho_0$	$U_\Lambda(0,\,\rho)$ (MeV)	Notes
0.5	–25 to –28	BHF/Chiral, with/without 3BF
1.0	–27 to –33	Empirical/ab initio
2.0	0 to +20	Onset of repulsion
3.0	+30 to +80	Strong repulsion at high $\rho$

The momentum dependence is moderate up to $k_\Lambda \sim 1.5$ fm$^{-1}$, with $U_\Lambda$ rising toward zero for high $k_\Lambda$ (Jinno et al., 4 Feb 2026, Inoue et al., 2016, Jinno et al., 27 Aug 2025). Momentum-dependent parametrizations, e.g.,

$$U_\Lambda(\rho, k) = a\,u + b\,u^{4/3} + c\,u^{5/3} + \frac{C}{\rho_0} \int d^3k'\,f(x,k')\,[1+((k-k')/\mu)^2]^{-1}$$

where $u \equiv \rho/\rho_0$, are routinely employed in transport and hydrodynamics codes (Jinno et al., 27 Aug 2025).

3. Empirical Extraction and Optical Potentials

Global fits to $\Lambda$ 1$s$ and 1$p$ binding energies across the periodic table using density-functional or optical-model approaches yield Woods–Saxon-like central potentials: $$V_\Lambda(\rho) = D_\Lambda^{(2)}\,\frac{\rho}{\rho_0} + D_\Lambda^{(3)}\,\left(\frac{\rho}{\rho_0}\right)^2$$ with $D_\Lambda^{(2)} = -38.6 \pm 0.8$ MeV, $D_\Lambda^{(3)} = +11.3 \pm 1.4$ MeV, so that $D_\Lambda(\rho_0) \approx -27.3 \pm 0.6$ MeV (Friedman et al., 2023). Here, the quadratic term encodes short-range, density-driven three-body repulsion; it dominates at high $\rho$, driving the potential repulsive and stiffening the equation of state.

In finite nuclei, typical Woods–Saxon parameters for $U_\Lambda(r)$ are depth $U_0 \sim 30$–$40$ MeV, radius $R \sim 1.1\,A^{1/3}$ fm, and diffuseness $a \sim 0.6$ fm (Vidana, 2016). Shell-structure models based on $u(3)\times u(2)$ dynamical symmetry recover similar level spacing and empirical gross features (Fortunato et al., 2016).

Direct reaction observables, such as scattering cross-sections, angular distributions, and rapidity spectra, are sensitive to $U_\Lambda$. In transport models, systematically varying $U_\Lambda(\rho_0)$ from $-50$ to $-20$ MeV at fixed beam energy modifies all observables, establishing experimentally testable signatures for potential extraction (Yong, 2024).

4. Ab Initio Approaches: Chiral EFT, Lattice QCD, and Three-Body Effects

Microscopic treatments based on chiral SU(3) EFT up to NLO/N$^2$LO, including full $G$-matrix summation, consistently generate $U_\Lambda(\rho_0,0) = -33$ to $-43$ MeV from two-body YN forces, but these overbind $\Lambda$ in hypernuclei (Jinno et al., 4 Feb 2026, Jinno et al., 27 Aug 2025). Inclusion of leading-order $\Lambda NN$ three-body forces, normal-ordered into effective two-body terms, supplies $\sim 10$–$15$ MeV repulsion at $\rho_0$ (Jinno et al., 27 Aug 2025, Haidenbauer et al., 2016, Kohno, 2018, Jinno et al., 16 Jan 2025). This yields net agreement with empirical $U_\Lambda$ and is essential to resolve the "hyperon puzzle"—the question of how massive neutron stars avoid collapse in the presence of softening by hyperons.

Ab initio lattice QCD potentials, processed via the HAL QCD method and embedded in BHF theory, give $U_\Lambda(\rho_0,0) = -33$ MeV without model-dependent phenomenology (Inoue et al., 2016). The momentum dependence from lattice data is parametrized as

$$U_\Lambda(\rho_0,k) \approx -33~\mathrm{MeV}\,\exp{\left[-(k/1.1~\mathrm{fm}^{-1})^2\right]}$$

valid up to $k \sim 2$ fm$^{-1}$.

5. Finite Nuclei, Spin–Orbit Splitting, and Spectroscopy

The $\Lambda$ single-particle potential in finite hypernuclei determines the spacings and quantum numbers of observed $\Lambda$ levels. Key features include:

• Small Spin–Orbit Splitting: Both empirical and theoretical studies (from mean-field, relativistic, and shell-model approaches) confirm a substantially weaker $\Lambda$N spin–orbit coupling compared to nucleons, yielding $p_{1/2}$–$p_{3/2}$ splittings of $\sim 0.1-0.3$ MeV (Vidana, 2016, Ding et al., 2023, Veselý et al., 2016).
• Shell Structure: Modern mean-field calculations (DD-RMF with density-dependent coupling, Skyrme-folded approaches) reproduce systematic variations in single-particle energies and radial potentials, reaching good agreement with experiment for level ordering and binding energies (Ding et al., 2023, Vidana, 2016, Fortunato et al., 2016).
• Correlation Strength: $Z$-factors for $\Lambda$ levels in finite nuclei are large (0.85–0.98), indicating much weaker correlations for hyperons relative to nucleons, consistent with infinite-matter results (Vidana, 2016).

6. Femtoscopy and Light Systems: $\Lambda$–$\alpha$ and Few-Body Approaches

In $^5_\Lambda$He and light systems, the $\Lambda$ single-particle potential is directly probed via folding approaches and femtoscopic correlation measurements (Jinno et al., 2024, Oo et al., 2020). Gaussian-type, Skyrme-folded, and microscopic G-matrix potentials (e.g., Isle, SG, Chi3) all reproduce the empirical separation energy ($B_\Lambda \approx 3.12$ MeV), but exhibit substantial variation in short-range repulsion, affecting high-momentum correlation functions. Overbinding in fully microscopic (separable NSC97f) models points to the critical role of short-range repulsion in reproducing the correct mean field (Oo et al., 2020).

7. Phenomenological Implications and Astrophysical Relevance

A repulsive or even mildly attractive $U_\Lambda(\rho,0)$ at high density ($\rho \gtrsim 2\rho_0$) is phenomenologically essential to delay or suppress $\Lambda$ appearance in neutron-star cores, thus ensuring a sufficiently stiff equation of state to support $2\,M_\odot$ stars (Jinno et al., 16 Jan 2025, Jinno et al., 27 Aug 2025, Haidenbauer et al., 2016, Friedman et al., 2023). This result is robust across chiral EFT, QCD-based, and phenomenological approaches. Heavy-ion observables (directed and elliptic flow of $\Lambda$, $\Sigma$) further constrain the momentum dependence of $U_\Lambda(p,\rho)$, especially for $k \gtrsim 1\,\mathrm{fm}^{-1}$ (Jinno et al., 27 Aug 2025, Jinno et al., 16 Jan 2025, Yong, 2024).

8. Summary Table: Representative Values and Parametrizations

Framework / Model	$U_\Lambda(\rho_0,0)$ (MeV)	High-Density Behavior	Momentum Dependence	Reference
Ab initio BHF (NLO chiral)	–33 to –43	$U_\Lambda(3\rho_0) = +30$ to $+80$	$\sim$ –40 MeV $\to$ 0 for $k\sim 2$ fm$^{-1}$	(Jinno et al., 4 Feb 2026, Jinno et al., 27 Aug 2025)
Chiral BHF + $\Lambda NN$	–30 (calibrated)	Strongly repulsive for $\rho \gtrsim 2\rho_0$	Significant at high $p$; influences flow	(Jinno et al., 16 Jan 2025, Haidenbauer et al., 2016)
Lattice QCD + BHF	–33	Not computed above $2\rho_0$	$\exp(-[k/1.1]^2)$ fm$^{-1}$ scaling	(Inoue et al., 2016)
Optical / Phenomenological	–27.3 ± 0.6	Repulsive for $\rho \gtrsim 3\rho_0$	N/A	(Friedman et al., 2023)
Finite Nuclei (Woods–Saxon fit)	12–35 (A=5–209)	N/A	Weak; levels as in experiment	(Vidana, 2016)

9. Theoretical Uncertainties

Principal theoretical uncertainties stem from:

• Cutoff and regulator dependence in chiral EFT ($\pm$10 MeV at $\rho_0$, $\pm$60 MeV at $3\rho_0$) (Jinno et al., 4 Feb 2026, Jinno et al., 27 Aug 2025)
• Three-body LECs ($c_D^\Lambda$, $c_E^\Lambda$) variation shifts high-density $U_\Lambda$ by tens of MeV (Jinno et al., 27 Aug 2025)
• Model assumptions in fitting optical-model or folding potentials, with error matrix degeneracy in two- vs three-body contributions (Friedman et al., 2023)
• Truncation of partial waves and omission of YNN forces at lower order (Inoue et al., 2016)

Empirical constraints from hypernuclear spectroscopy, heavy-ion reaction data, and neutron-star masses provide critical benchmarks to limit this uncertainty.

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In conclusion, the $\Lambda$ single-particle potential $U_\Lambda(p, \rho)$ is now quantitatively established at nuclear-matter density and well understood in its density and momentum evolution, supported by both first-principles and phenomenological data. Its accurate characterization remains foundational for hypernuclear structure theory and neutron-star astrophysics.