Universality of many-body two-nucleon momentum distributions: Correlated nucleon spectral function of complex nuclei

Background: The nuclear spectral function is a fundamental quantity that describes the mean-field and short-range correlation dynamics of nucleons embedded in the nuclear medium; its knowledge is a prerequisite for the interpretation of various electroweak scattering processes off nuclear targets aimed at providing fundamental information on strong and weak interactions. Whereas in the case of the three-nucleon and, partly, the four-nucleon systems, the spectral function can be calculated ab initio within a nonrelativistic many-body Schroedinger approach, in the case of complex nuclei only models of the correlated, high-momentum part of the spectral function are available so far.

Purpose: The purpose of this paper is to present a new approach such that the spectral function for a specific nucleus can be obtained from a reliable many-body calculation based upon realistic nucleon-nucleon interactions, thus avoiding approximations leading to adjustable parameters.

Methods: The expectation value of the nuclear many-body Hamiltonian, containing realistic nucleon-nucleon interaction of the Argonne family, is evaluated variationally by a normalization-conserving linked-cluster expansion and the resulting many-body correlated wave functions are used to calculate the one-nucleon and the two-nucleon momentum distributions; by analyzing the high-momentum behavior of the latter, the spectral function can be expressed in terms of a transparent convolution formula involving the relative and center-of-mass (c.m.) momentum distributions in specific regions of removal energy $E$ and momentum $k$.

Results: It is found that as a consequence of the factorization of the many-body wave functions at short internucleon separations, the high-momentum behavior of the two-nucleon momentum distributions in $A=3,4,12,16,40$ nuclei factorizes, at proper values of the relative and c.m. momenta, into the c.m. and relative momentum distributions, with the latter exhibiting a universal $A$-independent character. By exploiting the factorization property, it is found that the correlated part of the spectral function can be expressed in terms of a convolution formula depending upon the many-body relative and c.m. momentum distributions of a nucleon pair.

Conclusions: The obtained convolution spectral function of the three-nucleon systems, featuring both two-and three-nucleon short-range correlations, perfectly agrees in a wide range of momentum and removal energy with the ab initio spectral function, whereas in the case of complex nuclei the integral of the obtained spectral functions (the momentum sum rule) reproduces with high accuracy the high-momentum part of the one-nucleon momentum distribution, obtained independently from the Fourier transform of the nondiagonal one-body density matrix. Thus, the convolution spectral function we have obtained appears to indeed be a realistic microscopic, parameter-free quantity governed by the features of the underlying two-nucleon interactions.

Figure 1

(Top) The two-body density ${\rho }_n^{\left(2\right)}(r=|{\mathbf{r}}_1-{\mathbf{r}}_2\left|\right)$ obtained in the variational NCLCE calculation of Ref. [33] performed with the first six components of the Argonne $V{8}^{\prime }NN$ interaction [Eq. (24)] corresponding to the values of the ground-state energy and radius listed in Table 1. (Bottom) The same as in the top panel but in the case of the calculation of Ref. [26] performed with the AV18 NN interaction plus $\mathrm{UIX}$ 3N interaction.

Figure 2

The proton ($n_p=n_n$) one-nucleon momentum distribution of ${}^4\text{He}$ and ${}^{16}\text{O}$ obtained in Ref. [26] within the cluster variational Monte Carlo (CVMC) and in Ref. [33] within the NCLCE at the lowest order.

Figure 3

The momentum distributions of ${}^{16}\mathrm{O}$ calculated with different many-body approaches and similar realistic NN interactions: the cluster variational Monte Carlo (CVMC) results of Ref. [26] (full line); the normalization conserving linked cluster expansion (NCLCE) Ref. [33] (triangles); the fermion hypernetted chain method (FHNC) of Ref. [29] with $\mathrm{V}{8}^{\prime }$ interaction (squares); the two-nucleon correlation model (CS) of Ref. [8] (asterisks). The full dots represent the momentum distributions obtained by integrating the SF obtained within the nuclear matter local density approximation (LDA) [6, 9].

Figure 4

The quantity $100\frac{\mathrm{\Delta }n\left(k\right)}{n^{\mathrm{VMCV}}}$ [Eq. (37)], i.e., the percent deviation of the microscopic calculations of the momentum distributions of ${}^{16}\mathrm{O}$ shown in Fig. 3 taking the CVMC results of Ref. [26] as the reference results. LDA, Refs. [6, 9]; CS, Ref. [8]; FHNC, Ref. [29]; NCLCE, Ref. [33].

Figure 5

The proton-neutron center-of-mass (c.m.) momentum distributions [Eq. (39)] in ${}^4\mathrm{He}, {}^{12}\mathrm{C}$, and ${}^{40}\mathrm{Ca}$ calculated within microscopic many-body approaches. NCLCE, Ref. [19]; VMC, Ref. [24]; CS, Ref. [8].

Figure 6

The $pn$ and $pp$ relative momentum distributions [Eq. (40)] in ${}^4\mathrm{He}$ and ${}^{12}\text{C}$ calculated by the NCLCE (lines) [19] and the VMC (symbols) [24].

Figure 7

The $pn$ and $pp$ two-nucleon momentum distributions in ${}^4\mathrm{He}, n^{pn}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.},\theta )$, obtained in Ref. [19] in correspondence with 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 increases 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. (43).

Figure 8

The $pn$ and $pp$ two-nucleon momentum distributions $n^{pN}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.},\theta =0)$ for ${}^4\mathrm{He}$ (symbols) compared with the results of Eq. (44) (lines), where for $pn$ and $pp$ Eqs. (45) and  (48), respectively, have been used.

Figure 9

The same as in Fig. 8 but for ${}^{40}\mathrm{Ca}$.

Figure 10

The ab initio proton and neutron SF of ${}^3\mathrm{He}$ from Ref. [3] (red dots) compared with the convolution SF [Eq. (50), full line] obtained taking into account the constraint [Eq. (43)] on the value of $k_{\mathrm{rel}}^{-}$, which guaranties that the convolution formula includes indeed only the factorization region.

Figure 11

The SF of ${}^4\mathrm{He}, {}^{12}\mathrm{C}, {}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$ calculated with the convolution SF [Eq. (50)]. The dashed and dot-dashed lines represent, respectively, the $pn$ and the $pp$ SRC contributions.

Figure 12

The convolution spectral of ${}^{12}\mathrm{C}$ [Eq. (50)] and ${}^{16}\mathrm{O}$ (full line) compared with the effective convolution formula from Ref. [8] (dashed line).

Figure 13

A three-dimensional (3D) figure of the total SF [Eq. (49)] of ${}^{12}C$ illustrating the mean-field and SRC contributions: $P_{\mathrm{MF}}^{N_1}(k,E)$ (shown in red) is located in the region of removal energy $E\le 50\text{MeV}$ where the contribution from the $\ell =1$ and $\ell =0$ shells can be identified, whereas $P_{\mathrm{SRC}}^{N_1}(k,E)$ (shown in blue) completely exhausts the removal energy region with $E\ge 50\text{MeV}$. The two contributions from the $s$ and $p$ shells correspond to Woods-Saxon single-particle wave functions with spectroscopic factors equal to 0.9.

Figure 14

The SRC momentum sum rule $n_{\mathrm{SRC}}\left(k\right)\equiv n_1\left(k\right)={\int }_0^{\infty }P(k,E^{*})d{E}^{*}$ in ${}^4\mathrm{He}$ and ${}^{16}\mathrm{O}$. The full line represents the total momentum distribution obtained in Ref. [33] with the dashed and dot-dashed curves corresponding to the mean-field and SRC contributions, respectively. The full dots represent the SRC momentum distribution obtained by integrating the SRC convolution SF. It can be seen that the momentum sum rule is exactly satisfied by the convolution formula.

Figure 15

The convergence of the momentum sum rule $n_{\mathrm{SRC}}\left(k\right)\equiv n_1\left(k\right)={\int }_0^{E^{+}}P(k,E^{*})d{E}^{*}$. The partial momentum sum rule corresponding to increasing value of $E^{+}$. It can be seen that in order to obtain the correct momentum distributions (full line) in the region $k\ge 4{\text{fm}}^{-1}$ it is necessary to integrate the SF up to $E^{+}\simeq 400\text{MeV}$. Full, dashed, and dot-dashed curves as in Fig. 14.