Short-range correlations in nuclei with similarity renormalization group transformations

Background: Realistic nucleon-nucleon interactions induce short-range correlations in nuclei. To solve the many-body problem unitary transformations like the similarity renormalization group (SRG) are often used to soften the interactions.

Purpose: Two-body densities can be used to illustrate how the SRG eliminates short-range correlations in the wave function. The short-range information can however be recovered by transforming the density operators.

Method: The many-body problem is solved for ${}^4\mathrm{He}$ in the no core shell model (NCSM) with SRG transformed $\mathrm{AV}8{}^{\prime }$ and chiral N3LO interactions. The NCSM wave functions are used to calculate two-body densities with bare and SRG transformed density operators in two-body approximation.

Results: The two-body momentum distributions for $\mathrm{AV}8{}^{\prime }$ and N3LO have similar high-momentum components up to relative momenta of about $2.5{\text{fm}}^{-1}$, dominated by tensor correlations, but differ in their behavior at higher relative momenta. The contributions of many-body correlations are small for pairs with vanishing pair momentum but not negligible for the momentum distributions integrated over all pair momenta. Many-body correlations are induced by the strong tensor force and lead to a reshuffling of pairs between different spin-isospin channels.

Conclusions: When using the SRG it is essential to use transformed operators for observables sensitive to short-range physics. Back-to-back pairs with vanishing pair momentum are the best tool to study short-range correlations.

Figure 1

Contributions from the different $S,T$ channels to the total energy of ${}^4\mathrm{He}$ as a function of the flow parameter $\alpha$ for the $\mathrm{AV}8{}^{\prime }$ interaction (solid) and the N3LO interaction (dashed). The total energy ${\tilde{E}}_{\mathrm{tot}}$ (black lines) includes the Coulomb energy.

Figure 2

$\mathrm{AV}8{}^{\prime }$ (left) and N3LO (right) two-body densities in coordinate space calculated with bare (top) and SRG transformed density operators (bottom). $\alpha$ in units of ${\text{fm}}^4$. See also Fig. 1 in Ref. [43] for the $\mathrm{AV}8{}^{\prime }$ two-body densities in the different $S,T$ channels.

Figure 3

$\mathrm{AV}8{}^{\prime }$ (left) and N3LO (right) two-body densities in momentum space calculated with bare (top) and SRG transformed density operators (bottom). $\alpha$ in units of ${\text{fm}}^4$. See also Fig. 2 in Ref. [43] for the $\mathrm{AV}8{}^{\prime }$ two-body densities in the different $S,T$ channels.

Figure 4

$\mathrm{AV}8{}^{\prime }$ (top) and N3LO (bottom) relative momentum distributions in the different spin-isospin channels obtained with SRG transformed density operators. $\alpha$ in units of ${\text{fm}}^4$.

Figure 5

$\mathrm{AV}8{}^{\prime }$ (top) and N3LO (bottom) relative momentum distributions for pairs with vanishing pair momentum $K=0$ in the different spin-isospin channels obtained with SRG transformed density operators.

Figure 6

Momentum distributions for the bare $\mathrm{AV}8{}^{\prime }$ and N3LO interactions in the $S,T=1,0$ channel decomposed into contributions from pairs with relative orbital angular momentum $l=0,2,4$. For all pairs (top) and for pairs with vanishing pair momentum (bottom).

Figure 7

Total, $l=0$ and $l=2$ momentum distributions in the deuteron $\left(S,T=1,0\right)$ for the $\mathrm{AV}8{}^{\prime }$ and the N3LO interactions.

Figure 8

Relative probability to find pairs in the different spin-isospin channels as a function of relative momentum. For all pairs (top) or only for pairs with pair momentum $K=0$ (bottom).

Figure 9

Relative probability to find $pn$ or $pp$ pairs with pair momentum $K=0$ as a function of relative momentum.