Hypernuclear no-core shell model

We extend the no-core shell model (NCSM) methodology to incorporate strangeness degrees of freedom and apply it to single-$\mathrm{\Lambda }$ hypernuclei. After discussing the transformation of the hyperon-nucleon ($YN$) interaction into the harmonic-oscillator (HO) basis and the similarity renormalization group transformation applied to it to improve model-space convergence, we present two complementary formulations of the NCSM, one that uses relative Jacobi coordinates and symmetry-adapted basis states to fully exploit the symmetries of the hypernuclear Hamiltonian and one working in a Slater determinant basis of HO states where antisymmetrization and computation of matrix elements is simple and to which an importance-truncation scheme can be applied. For the Jacobi-coordinate formulation, we give an iterative procedure for the construction of the antisymmetric basis for arbitrary particle number and present the formulas used to embed two- and three-baryon interactions into the many-body space. For the Slater-determinant formulation, we discuss the conversion of the $YN$ interaction matrix elements from relative to single-particle coordinates, the importance-truncation scheme that tailors the model space to the description of the low-lying spectrum, and the role of the redundant center-of-mass degrees of freedom. We conclude with a validation of both formulations in the four-body system, giving converged ground-state energies for a chiral Hamiltonian, and present a short survey of the $A\le 7$ hyperhelium isotopes.

Figure 1

Momentum-space matrix elements of the $Q=0, \mathcal{S}=-1 {}^1\mathrm{S}_0$ partial wave for the bare and evolved leading-order $YN$ interaction [44] with a regulator cutoff of $\mathrm{\Lambda }=600\mathrm{MeV}/c$. The particle-combinations are coupled by the transition matrix elements in the bottom three rows and have to be evolved simultaneously. The low-momentum matrix elements of the ${\mathrm{\Sigma }}^{-}p$ channel (third row) are dominated by the Coulomb interaction.

Figure 2

Illustration of the Jacobi coordinates used for (a) antisymmetrization and embedding of the (b) $YN$, (c) $NN$, and (d) $NNN$ interactions.

Figure 3

Dimension of the full NCSM model space for natural parity and $M_J=\{0,1/2\}$ as function of $N_{\text{max}}$ for the two hypernuclei ${}_{\mathrm{\Lambda }}{}^5\mathrm{He}$ and ${}_{\mathrm{\Lambda }}{}^7\mathrm{He}$, compared to their nucleonic parents and the nuclei obtained by replacing the hyperon with a neutron.

Figure 4

Threshold extrapolation of the absolute ground-state energy and the excitation energy of the second excited state in ${}_{\mathrm{\Lambda }}{}^7\mathrm{He}$. The computed values (blue dots) are shown together with the best-fit polynomial (red line) and the uncertainty due to the unknown functional form, approximated by a set of different fit polynomials (gray band; see text for details). The regulator cutoff of the $YN$ interaction is $\mathrm{\Lambda }=700\mathrm{MeV}/c$.

Figure 5

Ground-state and excitation energy of ${}_{\mathrm{\Lambda }}{}^4\mathrm{He}$ for the (a) bare, (b) $\mathrm{SRG}{\left(3\right)}_N$ evolved to $\alpha =0.04{\mathrm{fm}}^4$ (red triangles) and $\alpha =0.08{\mathrm{fm}}^4$ (blue circles), and (c) $\mathrm{SRG}{\left(3\right)}_N+\mathrm{SRG}{\left(2\right)}_Y$ Hamiltonian, obtained from a J-NCSM calculation. The regulator cutoff is $\mathrm{\Lambda }=600\mathrm{MeV}/c$, and the basis frequency is $\hslash \mathrm{\Omega }=20\mathrm{MeV}$ for the evolved Hamiltonians and $\hslash \mathrm{\Omega }=30\mathrm{MeV}$ for the bare one. The crosses denote the IT-NCSM results for both flow parameters, and $E_{\infty }$ is the energy extrapolated to $N_{\text{max}}\to \infty$.

Figure 6

Absolute ground-state energy of (a) ${}^4\mathrm{He}$ and (b) its daughter hypernucleus ${}_{\mathrm{\Lambda }}{}^5\mathrm{He}$. The oscillator frequency is $\hslash \mathrm{\Omega }=20\mathrm{MeV}$. We use the $\mathrm{SRG}{\left(3\right)}_N$ Hamiltonian with a flow parameter of ${\alpha }_N=0.08{\mathrm{fm}}^4$ in the nucleonic sector and no SRG evolution, i.e., ${\alpha }_Y=0{\mathrm{fm}}^4$, of the YN interaction. The regulator cutoffs employed are $\mathrm{\Lambda }=700\mathrm{MeV}/c$ (solid lines) and $\mathrm{\Lambda }=600\mathrm{MeV}/c$ (dashed lines). Experimental data from Refs. [3, 63].

Figure 7

Absolute and excitation energies of low-lying states in (a) ${}^5\mathrm{He}$ and (b) its daughter hypernucleus ${}_{\mathrm{\Lambda }}{}^6\mathrm{He}$. Parameters are the same as in Fig. 6. Vertical lines mark extrapolation uncertainties; experimental data taken from Refs. [3, 63].

Figure 8

Absolute and excitation energies of low-lying states in (a) ${}^6\mathrm{He}$ and (b) its daughter hypernucleus ${}_{\mathrm{\Lambda }}{}^7\mathrm{He}$. Parameters are the same as in Fig. 6. Vertical lines mark extrapolation uncertainties; experimental data taken from Refs. [7, 63].