II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

On the role of three-particle interactions in nuclear matter†
W. Bentz∗1 and I. C. Cloët∗2
In a previous publication1) we discussed an interesting relation( between the) skewness J of nuclear matter(J = 27ρ3 d3 EA /dρ3 , where ρ is the baryon density
and EA the energy per nucleon in isospin symmetricnuclear matter) and the isoscalar three-particle interaction parameters. In this paper, we wish to discuss
an equally interesting relation between the slope pas
rameter L of the symmetry energy (L = 3ρ dadρ , whereas ≃ 32 MeV is the symmetry energy) and the isovector three-particle interaction parameters.
We extend Landau’s basic formula2) for the variation of the energy density of nuclear matter to include the third order term, which involves the spinaveraged three-particle forward scattering amplitudeh(τ1 τ2 τ3 ) (⃗k1 , ⃗k2 , ⃗k3 ). Here τi = (p, n), and h is symmetric under simultaneous interchanges of the momentum
variables ⃗ki and the isospin variables τi . Taking ﬁnally
the isospin symmetric limit, we can derive the following
relations for J and L in terms of the incompressibility
K and the symmetry energy as :tt

Here F0 , F0′ , G0 , G′0 are the dimensionless ℓ = 0 twoparticle Landau-Migdal parameters, as deﬁned for ex(2pc)
is positive
ample in Ref. 2). While the isoscalar H0deﬁnite and of the order of unity or even larger, depending mainly on the magnitude of G′0 , the isovector′(2pc)
H0
is negative and small compared to unity formost of the published sets of Landau-Migdal parameters. Because the p-wave term H1 is suppressed bylarge factors,1) this simple estimate makes it plausiblethat the three-body interactions give a large positive
contribution to J, and a small negative contribution
to L. To obtain more quantitative results, it would beinteresting to apply the Faddeev method in the framework of eﬀective ﬁeld theories for nuclear matter.

∆M ∗ = M ∗(p) − M ∗(n) on the isovector density ρ(3) =ρ(p) − ρ(n) . The dimensionless isoscalar and isovectorthree-particle interaction parameters()()
2pF M ∗
2pF M ∗′ρh,H=ρh′ℓ
Hℓ =ℓℓπ2π2are the ℓ = 0, 1 moments of the isoscalar (hℓ =(ppp)(ppn)(ppp)
1+ 3hℓ)) and isovector (h′ℓ = 14 (hℓ−4 (hℓ(ppn)hℓ)) combinations of the 3-particle forward scattering amplitude at the Fermi surface.
By using empirical information, it was shown inRef. 1) that the above expression for J requires a large
Mpositive 3-particle term M∗ (H0 − H1 ) > 1.24. On theother hand, if we use the canonical value as = 32 MeV∗2ttogether with µ ≃ 0.27, which is the central value ofthe empirical range µ = 0.27±0.25 reported in Ref. 3),
the sum of the ﬁrst two terms in [. . . ] in the expression for L is ∼ 0.6, almost independent of M ∗ within
the empirical range 0.7 < M ∗ /M < 1. The empirical range of the slope parameter3) L = 59
( ±′ 16 1MeV)
M
then implies that the 3-particle term M
H 0 − 3 H1∗
is negative, with a magnitude smaller than unity.
Theoretically the three-particle amplitudes should
be calculated from the Faddeev equation, which is illustrated by Fig. 1. The driving term, which we callthe “2-particle correlation (2pc) term,” can be easily
estimated by using eﬀective contact interactions of theLandau-Migdal type. Restricting the calculation to swaves (ℓ = 0) for simplicity gives the analytic results)′′
ln 2 ( 2(2pc)
F0 + 3F02 + 3G20 + 9G02 ,=H04()′′1 ′2
ln 2 1 2 4(2pc)
2′2′
F + F0 F0 − F0 + G0 + 4G0 G0 − G0 .
H0=4 3 0 3
3Here pF is the Fermi momentum, M the free nucleon∗mass, M ∗ the Landau eﬀective mass, ∂M∂pF refers to the∗momentum dependence() of M at the Fermi surface,∂∆M ∗and µ = ρ ∂ρ(3)expresses the dependence ofM∗1...
tFig. 1. First two terms in the Faddeev series. Circles represent two-body t-matrices.

9p2
J = −9K + FM )[(]8 M
4 M pF ∂M ∗ M
× −3++(H0 −H1 ) ,−3 M∗
3 M ∗2 ∂pF M ∗p2
L = 3as − F
2M[()()2()]2 M
M
1M′× 1−− ∗ H 0 − H1+µ.
3 M∗
M∗
M
3†tReferences

1) W. Bentz, I. C. Cloët, Phys. Rev. C 100, 014303 (2019).
2) J. Negele, H. Orland, Quantum Many-particle systems(Springer, 1980).
3) B. -A. Li, B. -J. Cai, L. -W. Chen, J. Xu, Prog. Part.
Nucl. Phys. 99, 29 (2018).

Condensed from an article by W. Bentz and I. C. Cloët, tobe published in Phys. Rev. C (2021)
Department of Physics, Tokai University
Physics Division, Argonne National Laboratory- 51 -1