Momentum-space evolution of chiral three-nucleon forces

A framework to evolve three-nucleon ($3\mathrm{N}$) forces in a plane-wave basis with the similarity renormalization group (SRG) is presented and applied to consistent interactions derived from chiral effective field theory at next-to-next-to-leading order. We demonstrate the unitarity of the SRG transformation, show the decoupling of low and high momenta, and present the first investigation of universality in chiral $3\mathrm{N}$ forces at low resolution scales. The momentum-space-evolved $3\mathrm{N}$ forces are consistent and can be directly combined with the standard SRG-evolved two-nucleon (NN) interactions for ab initio calculations of nuclear structure and reactions.

Figure 1

Ground-state energy of ${}^3$H as a function of the flow parameter $\lambda =s^{-1/4}$ for different chiral interactions at N${}^2$LO, which are labeled by the value of the cutoffs $\Lambda /\tilde{\Lambda }$ (see Ref. [19]). The solid lines show the results for $\Lambda /\tilde{\Lambda }=550/600$ MeV separately for NN-only, $\mathit{NN}+3\mathrm{N}$-induced, and $\mathit{NN}+3\mathrm{N}$-full (see Ref. [14]). The dashed lines show the $\mathit{NN}+3\mathrm{N}$-full results for the other given values of $\Lambda /\tilde{\Lambda }$. Only $np$ NN interactions have been used.

Figure 2

Ground-state energy of ${}^3$H as a function of the flow parameter $\lambda =s^{-1/4}$ for different SRG model space sizes $N_{\alpha }$ and a fixed model space for the initial Hamiltonian $N_{\alpha }^0=5$ (for a detailed definition of the $3\mathrm{N}$ channels, see Ref. [17]). The dotted line shows the energy at $\lambda =\infty$. Only $np$ NN interactions have been used.

Figure 3

Contour plots of the evolved $3\mathrm{N}$ potential $\langle \xi \alpha =1\left|{\overline{V}}_{123}\right|{\xi }^{\prime }{\alpha }^{\prime }=1\rangle$ at the hyperangle $\theta =\pi /12$ (see Ref. [17] for definition of partial waves). The top panel shows the potential with $\Lambda /\tilde{\Lambda }=550/600$ MeV and the bottom panel $\Lambda /\tilde{\Lambda }=450/500$ MeV.

Figure 4

Matrix elements of the initial $3\mathrm{N}$ forces (dotted lines) compared to evolved $3\mathrm{N}$ forces (solid lines) at $\lambda =1.5{\mathrm{fm}}^{-1}$ for different interactions labeled by the values of the cutoffs $\Lambda /\tilde{\Lambda }$. We choose $p^{\prime }=1.0{\mathrm{fm}}^{-1}$, $q^{\prime }=1.25{\mathrm{fm}}^{-1}$, and $\alpha ={\alpha }^{\prime }=2$ (see Ref. [17]). In the left panel we fixed $q=1.5{\mathrm{fm}}^{-1}$ and in the right panel $p=0.75{\mathrm{fm}}^{-1}$. The shaded band marks the maximal variation between different evolved $3\mathrm{N}$ force matrix elements.