Hebeler EFT Baryon Phases

• The paper details a comprehensive EFT framework that models three-baryon interactions using chiral effective field theory and partial-wave analysis.
• The methodology extends traditional approaches by incorporating full tensor rank expansions, enabling accurate computation of partial-wave matrix elements.
• The phenomenological implementation via piece-wise polytropes provides practical insights into mass–radius relations and gravitational-wave signatures in compact stars.

The Hebeler-et al. EFT baryon phases constitute a set of phenomenological and microscopic equations of state (EoS) for dense baryonic matter, constructed using chiral effective field theory (EFT) and refined partial-wave analysis methods. These phases underpin modeling of three-baryon interactions (notably $\Lambda NN$) in hypernuclear and astrophysical contexts, and are leveraged as benchmark baryonic EoS in constructing hybrid compact-star models that transition to deconfined quark matter at high densities. The formalism originated with the work of Hebeler, Krebs, Epelbaum, Golak, and Skibiński ("Phys. Rev. C 91, 044001 (2015)"), which was subsequently extended to full tensor ranks by Kohno, Kamada, and Miyagawa (Kohno et al., 2022). The EoS have also been implemented phenomenologically (e.g., as piece-wise polytropes) in hybrid matter models to paper properties of neutron stars and possible quark matter cores (Fadafan et al., 2 Dec 2025).

1. Chiral Effective Field Theory and Three-Baryon Forces

In chiral EFT, low-energy baryonic interactions are systematically expanded in terms of pion and nucleon degrees of freedom, preserving chiral symmetry. Three-baryon forces (3BFs) enter at subleading orders and are crucial for describing dense baryonic matter and hypernuclei. The original Hebeler et al. approach formalized the projection of local chiral 3BFs onto a Jacobi momentum basis:

• The three-body operator $V_{3BF}(\vec{Q}_1, \vec{Q}_2, \vec{Q}_3)$, with $\vec{Q}_i = \vec{p}_i' - \vec{p}_i$ and $\sum \vec{Q}_i = 0$, is expressed in terms of pair $(\vec{p})$ and spectator $(\vec{q})$ momenta.
• Partial-wave states $|p\, q\, \alpha\rangle$ encode all relevant quantum numbers $(\ell_p, s_p, j_p; \ell_q, s_q, j_q; J, T)$.
• For a local 3BF, dummy integrations and expansions in Legendre polynomials allow for a reduction to double integrals and sums over angular-momentum indices.

This framework enabled efficient and accurate computation of partial-wave matrix elements for nuclear and hypernuclear structure calculations (Kohno et al., 2022).

2. Generalization to Arbitrary Spin–Momentum Rank

Kohno et al. generalized the Hebeler et al. method to incorporate higher spin–momentum tensor ranks ($K$), essential for capturing the full structure of three-body interactions:

• The partial-wave matrix elements are given by

$$V_{\alpha'\alpha}^{J;K}(p',q';p,q) = N_{\alpha'\alpha}^{J;K} \int dq\,q^2 \int dp\,p^2 \sum_{k,k'} P_{k'}(c_p)\,P_k(c_q) \sum_{j_p+j_p' = \ell_p'} \sum_{j_q+j_q' = \ell_q'} C\,V^{(K)}(p,q)\,,$$

with explicit recoupling coefficients, Legendre polynomials, and momentum factors.

• Each term in the interaction is organized according to tensor rank $K$ and momentum transfer channels $(\ell_a, \ell_b)$, with a complete set of Clebsch-Gordan, Wigner $6j$ and $9j$ symbols ensuring rotational invariance.
• The master formula circumvents the K=0 limitation of the original approach, permitting explicit evaluation of spin–momentum couplings up to $K=2$ (relevant for leading two-pion-exchange $\Lambda NN$ forces).

This comprehensive formalism allows for systematic calculation of all components of the three-baryon interaction, enabling high-precision studies of nuclear and hypernuclear systems (Kohno et al., 2022).

3. Phenomenological Implementation: Piece-wise Polytropes

Hebeler et al. constructed zero-temperature neutron-matter EoS families ("soft," "medium," "stiff") in a piece-wise polytropic form suitable for astrophysical modeling:

• The EoS is represented as

$$P_{\rm b}(\epsilon) = \begin{cases} K_1\,(\epsilon/\epsilon_0)^{\Gamma_1}, & \epsilon_0 \le \epsilon < \epsilon_1 \ K_2\,(\epsilon/\epsilon_1)^{\Gamma_2}, & \epsilon_1 \le \epsilon < \epsilon_2 \ K_3\,(\epsilon/\epsilon_2)^{\Gamma_3}, & \epsilon \ge \epsilon_2 \ \end{cases}$$

where $\epsilon$ is the energy density, matched at ($\epsilon_1$, $\epsilon_2$), and $K_i$ are set by continuity.

• The "medium" and "stiff" EoS, with larger $\Gamma_i$ in each segment for the "stiff" case, have found use as baryonic components in hybrid EoS for neutron stars.
• Implementation involves transforming $P_b(\epsilon)$ into the grand-canonical ensemble, facilitating comparison and matching with quark matter EoS in hybrid matter models.

These polytropic representations are standard in neutron-star structure modeling, and their parameters are informed by microscopic EFT calculations (Fadafan et al., 2 Dec 2025).

4. Applications in Hypernuclei and Compact Stars

The partial-wave expansion formalism is critical for calculating observables in hypernuclei and compact objects:

• In hypernuclear physics, the $\Lambda$–deuteron folding potential ($U_{\Lambda d}^{J_t}(q',q)$) is constructed by integrating three-body matrix elements with deuteron and $\Lambda$ wavefunctions, including spin–isospin recoupling and explicit isospin factors; this folding potential is essential for evaluating the $\Lambda NN$ three-baryon force's contribution to the hypertriton separation energy, found to be $100$–$200$ keV in the S-wave, comparable to experimental values (Kohno et al., 2022).
• For neutron stars and hybrid stars, the Hebeler EoS serve as the baryonic sector in matched EoS with deconfined quark matter, permitting analysis of phase transitions and their effect on observables such as the maximum mass and tidal deformability (Fadafan et al., 2 Dec 2025).

A plausible implication is that EFT-based baryonic phases, when sufficiently stiff, enable configurations with stable quark cores, affecting the observable mass–radius relation and gravitational-wave signatures in agreement with current pulsar mass constraints.

5. Hybrid Equations of State and Phase Transition Matching

In models of hybrid compact stars, baryonic (Hebeler) and quark phases are matched using standard thermodynamic criteria:

• The pressure matching (Maxwell construction) is imposed at the transition chemical potential $\mu_{\rm tr}$:

$P_{\rm b}(\mu_{\rm tr}) = P_{\rm q}(\mu_{\rm tr})$

• Discontinuity in the energy density leads to a "latent-heat" jump

$$\Delta \epsilon = \epsilon_{\rm q}(\mu_{\rm tr}) - \epsilon_{\rm b}(\mu_{\rm tr}) = \mu_{\rm tr} [n_{\rm q}(\mu_{\rm tr}) - n_{\rm b}(\mu_{\rm tr})]$$

• The transition is classified as "weakly first order" if $$\Delta \epsilon / \epsilon_{\rm tr} \lesssim {\cal O}(0.1)$$, a condition necessary for the stability of hybrid stars with quark cores.

In practical implementations, such as those using the IR-modified D3/D7 holographic model, only the "stiff" Hebeler EoS can accommodate a weak enough transition for stable hybrid stars. For the "medium" EoS, stability is lost as soon as the central energy density exceeds the transition value. The hybrid EoS then determines the input for Tolman–Oppenheimer–Volkoff equations, from which global stellar properties follow (Fadafan et al., 2 Dec 2025).

6. Mass–Radius Relations and Observational Implications

By integrating the Tolman–Oppenheimer–Volkoff system with the hybrid EoS,

• The maximum mass $M_{\rm max}$ and mass–radius curves are computed. For the stiff Hebeler EoS matched to a suitably tuned quark EoS, maximum masses in the range $1.90$–$2.17\,M_\odot$ are found, consistent with the masses of the heaviest observed pulsars.
• The tidal deformability $\Lambda$ is calculated using Love number boundary conditions; $\Lambda$ drops sharply (by $\sim 80\%$) when a quark core appears, which provides a distinguishing phenomenological signal. Pure baryonic stars typically overshoot the LIGO/Virgo tidal deformability constraints, but the presence of a quark core via a hybrid EoS yields values compatible with observational bounds.
• The equation of state parametrization and transition properties thus play a decisive role in the structure and gravitational-wave fingerprints of heavy compact objects.

A plausible implication is that the Hebeler-stiff baryon phase, as implemented in hybrid EoS, provides one of the minimal and well-controlled frameworks for deducing whether deconfined quark matter might reside in the centers of massive neutron stars (Fadafan et al., 2 Dec 2025).

7. Significance for Dense Baryonic and Hyperonic Matter

Understanding the Hebeler-et al. EFT baryon phases is essential for:

• Accurate predictions of hypernuclear spectra and hypertriton binding energies, where three-baryon forces must be accounted for at the same level as two-body interactions.
• Modeling the appearance of hyperons and possible phase transitions in neutron-star matter, which impacts the equation of state and the maximum mass of compact stars.
• Informing the construction and interpretation of hybrid matter equations of state relevant to gravitational-wave astrophysics and future multi-messenger constraints.

Further future work will likely focus on extending the formalism to more exotic components (e.g., other hyperons, higher strangeness), coupling to more sophisticated quark matter models, and refining the uncertainties in the EFT parameters to improve constraints from astronomical data (Kohno et al., 2022, Fadafan et al., 2 Dec 2025).