Microscopic nucleon spectral function for finite nuclei featuring two- and three-nucleon short-range correlations: The model versus ab initio calculations for three-nucleon systems

Background: Two-nucleon ($2N$) short-range correlations (SRC) in nuclei have been recently thoroughly investigated, both theoretically and experimentally and the study of three-nucleon ($3N$) SRC, which could provide important information on short-range hadronic structure, is underway. Novel theoretical ideas concerning $2N$ and $3N$ SRC are put forward in the present paper.

Purpose: The general features of a microscopic one-nucleon spectral function which includes the effects of both $2N$ and $3N$ SRC and its comparison with ab initio spectral functions of the three-nucleon systems are illustrated.

Methods: A microscopic and parameter-free one-nucleon spectral function expressed in terms of a convolution integral involving ab initio relative and center-of-mass (c.m.) momentum distributions of a $2N$ pair and aimed at describing two- and three-nucleon short-range correlations, is obtained by using: (i) the two-nucleon momentum distributions obtained within ab initio approaches based upon nucleon-nucleon interactions of the Argonne family; (ii) the exact relation between one- and two-nucleon momentum distributions; (iii) the fundamental property of factorization of the nuclear wave function at short internucleon ranges.

Results: The comparison between the ab initio spectral function of ${}^3\mathrm{He}$ and the one based upon the convolution integral shows that when the latter contains only two-nucleon short-range correlations the removal energy location of the peaks and the region around them exhibited by the ab initio spectral function are correctly predicted, unlike the case of the high and low removal energy tails; the inclusion of the effects of three-nucleon correlations brings the convolution model spectral function in much better agreement with the ab initio one; it is also found that whereas the three-nucleon short-range correlations dominate the high energy removal energy tail of the spectral function, their effects on the one-nucleon momentum distribution are almost one order of magnitude less than the effect of two-nucleon short-range correlations.

Conclusions: The convolution model of the spectral function of the three-nucleon systems featuring both two- and three-nucleon short-range correlations and correctly depending upon the ab initio two-nucleon relative and center-of-mass momentum distributions provides in the correlation region a satisfactory approximation of the spectral function in a wide range of momentum and removal energy. The extension of the model to complex nuclei is expected to provide a realistic microscopic parameter-free model of the spectral function, whose properties are therefore governed by the features of realistic two-nucleon interactions and the momentum distributions in a given nucleus.

Figure 1

The ab initio neutron spectral function of ${}^3\mathrm{He}$ [Eq. (14)] calculated in Ref. [15] using $3N$ wave functions corresponding to the RSC interaction [16] (a) and in Ref. [17] using $3N$ wave functions [18] corresponding to the AV18 [19] interaction (b). In both cases the full lines represent the PWIA (the proton-proton wave function ${\mathrm{\Psi }}_{pp}^{\mathbf{t}}$ in the final state is the exact solution of the the Hamiltonian which has been used to obtain the ground-state wave function), whereas the dot-dashed line in (a) and the dotted line in (b) correspond to the PWA (the final $pp$ state is approximated by a plane wave). The regions where the PWA practically coincides with the PWIA are clearly visible and correspond to high values of the momentum. In (b) $E_{\mathrm{rel}}=E+\left|E_3\right|, |E_3|$ being the binding energy of ${}^3\mathrm{He}$ and $E_{\mathrm{rel}}$ is the relative energy of the proton-proton pair in the continuum. Therefore the energy scales in the two Figures differ only by the small value $|E_3|=7.718\mathrm{MeV}$.

Figure 2

The ab initio neutron spectral function of ${}^3\mathrm{He}$ shown in Fig. 1 in correspondence of several values of the neutron momentum in PWA and PWIA, i.e., respectively, by disregarding (blue squares) and including (red triangles) the interaction in the $pp$ final state. It can be seen that at high values of the neutron momentum ($k\gtrsim 2.5{\mathrm{fm}}^{-1}$) the final state interaction in the the $pp$ essentially affects only the spectral function at small values of the excitation energy $E^{*}$ (in this and the following figures $E^{*}=E_{\mathrm{rel}}$).

Figure 3

(a) Pictorial representation of the kinematics of $2N$ SRC in nucleus $A$: the high momentum ${\mathbf{k}}_1\equiv \mathbf{k}\gtrsim 2.0{\mathrm{fm}}^{-1}$ of nucleon “1” is almost completely balanced by the momentum ${\mathbf{k}}_2\simeq -\mathbf{k}$ of the correlated partner nucleon “2”, with the residual system moving with low momentum $|{\mathbf{K}}_{A-2}|=|{\mathbf{k}}_1+{\mathbf{k}}_2|\lesssim 1.0{\mathrm{fm}}^{-1}$. Momentum conservation reads as follows: ${\sum }_1^A{\mathbf{k}}_i={\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{K}}_{A-2}=0$. In case of $A=3$ the $(A-2)$ nucleus is just a nucleon with momentum ${\mathbf{K}}_{A-2}={\mathbf{k}}_3$. (b) Pictorial representation of the kinematics of $3N$ SRC in nucleus $A$. The three nucleons have high momenta and low c.m. momentum which is balanced by the momentum of the system $A-3$. In the case of $A=3$ the configuration in (${\mathbf{b}}_{\mathbf{1}}$) can affect only the low removal energy part of the spectral function, whereas in the configuration (${\mathbf{b}}_{\mathbf{2}}$) also the high removal energy part can be affected.

Figure 4

The c.m. momentum distribution of the correlated proton-neutron pair in ${}^3\mathrm{He}$ calculated in Ref. [5] (full line) and in Ref. [3] (open dots) with ab initio wave functions corresponding to the AV18 interaction. The figure shows the separation into the soft (dashed line) and hard (dotted line) components. (Adapted from Ref. [5].)

Figure 5

The $pn$ two-nucleon momentum distributions in ${}^3\mathrm{He}, n^{pn}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.},\theta )$, obtained ab initio in Ref. [5] in correspondence of several values of $K_{\mathrm{c}.\mathrm{m}.}$ and two values of the angle $\theta$ between ${\mathbf{K}}_{\mathrm{c}.\mathrm{m}.}$ and ${\mathbf{k}}_{\mathrm{rel}}$. The region of $k_{\mathrm{rel}}$ where the value of $n^{pn}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.},\theta )$ is independent of the angle determines the region of factorization of the momentum distributions, i.e., $n^{pn}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.},\theta )\to n_{\mathrm{rel}}^{pn}\left(k_{\mathrm{rel}}\right)n_{\mathrm{c}.\mathrm{m}.}^{pn}\left(K_{\mathrm{c}.\mathrm{m}.}\right)$. It can be seen that the region of factorization starts at values of $k_{\mathrm{rel}}=k_{\mathrm{rel}}^{-}$, which increase with increasing values of $K_{\mathrm{c}.\mathrm{m}.}$, i.e., $k_{\mathrm{rel}}^{-}=k_{\mathrm{rel}}^{-}\left(K_{\mathrm{c}.\mathrm{m}.}\right)$; because of the dependence of $k_{\mathrm{rel}}^{-}$ upon $K_{\mathrm{c}.\mathrm{m}.}$, a constraint on the region of integration over $K_{\mathrm{c}.\mathrm{m}.}$ arises from Eq. (31). (Adapted from Ref. [5].)

Figure 6

Full line: the ab initio spectral function of the neutron in ${}^3\mathrm{He}$ in PWA corresponding to the AV18 interaction, shown in Fig. 2 by the full squares. Dot-dashed line: convolution model which includes $2N$ SRC only ($k_{\mathrm{rel}}>2{\mathrm{fm}}^{-1}, K_{\mathrm{c}.\mathrm{m}.}\le 1.0{\mathrm{fm}}^{-1}$); Dashed line: convolution model which includes both $2N$ ($k_{\mathrm{rel}}>2{\mathrm{fm}}^{-1}, K_{\mathrm{c}.\mathrm{m}.}\le 1.0{\mathrm{fm}}^{-1})$ and $3N$ ($k_{\mathrm{rel}}>2{\mathrm{fm}}^{-1}, K_{\mathrm{c}.\mathrm{m}.}>1.0{\mathrm{fm}}^{-1})$ SRC. Both dashed and dot-dashed lines include the constraint on the values of $K_{\mathrm{c}.\mathrm{m}.}$ imposed by the requirement of factorization [Eq. (31)]. Dotted line: convolution model of Ref. [8] which uses only the soft part of the c.m. momentum distribution without the constraint on the value of $K_{\mathrm{c}.\mathrm{m}.}$ [Eq. (31)]. The full dots in the case of $k=3.0{\mathrm{fm}}^{-1}$ denote the contribution from $3N$ SRC, i.e., the difference between the dashed and the dot-dashed curves.

Figure 7

The contributions to the neutron spectral function of the soft and hard parts of the c.m. momentum distributions shown in Fig. 4 in the case of $k=2.5{\mathrm{fm}}^{-1}$ (cf. Fig. 6) considering (Const.) and disregarding (NoConst.) the constraint on the value of $K_{\mathrm{c}.\mathrm{m}.}$ generated by Eq. (31).

Figure 8

The ab initio neutron momentum distribution in ${}^3\mathrm{He}$ [22] (full line) compared in the high momentum region with the distribution obtained from the momentum sum rule [Eq. (6)], i.e., by integrating the convolution model spectral function [Eq. (27)]. Dotted line: contribution from $3N$ SRC; Dashed line: contribution from $2N$ SRC; Full dots: the sum of $2N$ and $3N$ SRC contributions.