Similarity renormalization group evolution of three-nucleon forces in a hyperspherical momentum representation

A new framework for computing the similarity renormalization group (SRG) evolution of three-nucleon forces (3NF) in momentum representation is presented. The use of antisymmetric three-particle hyperspherical momentum states ensures unitary evolutions within certain basis truncations, much like antisymmetric harmonic oscillator SRG evolutions. Additionally, in each partial wave the ${\mathit{T}}_{\mathrm{rel}}$-SRG regulator is exactly represented, similar to recent 3NF momentum representation evolutions. Unitary equivalence is demonstrated for the triton using several chiral two- plus three-nucleon interactions. This method allows for a clean visualization of the evolution of the three-nucleon forces, which manifests the SRG decoupling pattern and low-momentum universality.

Figure 1

HH SRG evolution of the triton binding energy using a ${\mathrm{N}}^2\mathrm{LO}500/600\mathrm{MeV}$ interaction, including only the 2NF, the 2NF $+$ induced 3NF, and the 2NF $+$ full 3NF (see Ref. [8]). The dashed lines are for the other ${\mathrm{N}}^2\mathrm{LO}$ 2NFs + 3NFs (see [22, 23]).

Figure 2

SRG evolution of the triton probability distribution as a function of hypermomentum $Q$ for several different SRG $\lambda$s for the interactions in Fig. 1. These are plotted as bands that span the range of the wave functions from different initial 2N + 3N ${\mathrm{N}}^2\mathrm{LO}$ interactions.

Figure 3

Color contour plot of the 3NF (upper row) and 2NF (lower row) as a function of the $\lambda$ for the 550/600 MeV potential. The lowest antisymmetric HH partial wave is plotted as well as the linear combination of two-body partial waves that are embedded for this three-body partial wave.

Figure 4

Same as Fig. 3 but using the 600/700 MeV potential.

Figure 5

Contour plot of the integrand of triton 3NF expectation value, $I(Q,Q^{\prime })=\frac{M}{{\hslash }^2}{\Psi }_i\left(Q\right){\Psi }_j^{†}\left(Q^{\prime }\right){\left(Q{Q}^{\prime }\right)}^5{V}_{i,j}^{3NF}(Q,Q^{\prime })$. The upper row is the integrand for several initial N${}^2$LO chiral potentials, while the lower row is the integrand for the SRG evolved interactions.