Relations between three-particle interactions in nuclear matter to observable quantities

TL;DR

The paper addresses how three-particle interactions in nuclear matter influence observable bulk quantities by deriving model-independent relations between the slope parameters of the symmetry energy and incompressibility, $L$ and $J$, and the $ extit{ell}=0$ moments of in-medium three-nucleon forward scattering amplitudes. It uses the Landau-Migdal Fermi liquid framework to express these slopes in terms of dimensionless three-body parameters $H_\ell$ and $H'_\ell$, and then analyzes the physical content of the three-particle amplitude, decomposing it into retardation terms and a three-body kernel $K^{(3)}$, with a BBG-inspired ladder expansion up to third order revealing both two-body and higher-order three-body contributions. Semi-quantitative assessments with empirical inputs suggest $H_0>1$ (isoscalar) and $H'_0<0$ (isovector) with $|H'_0|<1$, while $H_1$ is subleading; the results imply that three-body cluster terms may be needed to account for the symmetry energy slope $L$. The work highlights the emergence of medium-induced four-body-like effects in the in-medium three-particle amplitude and establishes a concrete link between microscopic three-body dynamics and macroscopic nuclear observables, with implications for nuclei and neutron-star structure.

Abstract

In the first part of this paper, we use the framework of the Fermi liquid theory to derive model-independent relations between the slope parameters of the symmetry energy and of the incompressibility in nuclear matter to three-particle interaction parameters. Based on these relations, we present simple estimates and compare with the empirical information. In the second part, we discuss the general structure of the three-particle scattering amplitude in nuclear matter, and use methods similar to the Bethe-Brueckner-Goldstone theory to show how three-particle cluster diagrams emerge naturally in the Fermi liquid theory.

Figures (5)

• Figure 1: First two terms in the Faddeev series for the three-particle scattering amplitude $h$. Lines with arrows denote nucleons, and $t$ denotes the two-particle $t$-matrix.
• Figure 2: Graphical representation of the three-particle amplitude $\tilde{h}(1, 2, 3)$. The three-particle kernel $K^{(3)}$, defined in the main text, is irreducible in the particle-hole channel, and contains terms of second, third and higher orders in the two- particle $t$-matrix (denoted as $t$). The products of pole parts of propagators with the same momenta ($S_{\rm pole}^2(k)$ or $S_{\rm pole}^3(k)$) in these diagrams are defined as $S_p^2(k) + S_h^2(k)$ or $S_p^3(k) + S_h^3(k)$, without products of particle and hole parts.
• Figure 3: Skeleton diagrams for the self energy to first order (a) and to third order (b) in the ladder $t$-matrix. Note that there is no second order contribution. The two-particle $t$-matrix in the ladder approximation is denoted by a black dot.
• Figure 4: Contribution of third order in the ladder $t$-matrix to the three-particle amplitude $\tilde{h}$. The diagram (a) is the second (next-to-leading) term in the Faddeev series. $\lambda_{ijk}$ is unity for even permutations of (123) and zero otherwise, and an independent sum over ($i j k$) and ($l m n$) is implied. The diagrams (b) and (c) arise from the interaction between the three given particles and the particles in the Fermi sea, and do not appear in the usual Faddeev series. $\varepsilon_{ijk}$ is the usual antisymmetric tensor, and a sum over $i, j, k = 1, 2, 3$ is implied. The two-particle $t$-matrix in the ladder approximation is denoted by a black dot.
• Figure 5: Photo taken in summer 1963 in front of the Physics Building at Argonne National Laboratory. From left to right: A. W. Martin, A. Saperstein, K. C. Wali, H. J. Lipkin, M. Peshkin, W. D. McGlinn, F. Coester, J. Monahan, K. Hiida, A. M. Green, G. E. Brown, A. Arima, K. Tanaka, D. Kurath, M. A. Melvin, N. Rosenzweig, K. Ekstein, S. Tani, M. N. Hack, B. Kursunoglu, T. Sebe, R. A. Ferrell, D. R. Inglis, B. M. Udaonkar, S. Fallieros, S. P. Pandya. Reprinted with kind permission of Argonne National Laboratory.