Universality of nucleon-nucleon short-range correlations: The factorization property of the nuclear wave function, the relative and center-of-mass momentum distributions, and the nuclear contacts

Background: The two-nucleon momentum distributions of nucleons $N_1$ and $N_2$ in a nucleus $A, n_A^{N_1{N}_2}({\mathbf{k}}_{\mathrm{rel}},{\mathbf{K}}_{\mathrm{c}.\mathrm{m}.})$, is a relevant quantity that determines the probability of finding two nucleons with relative momentum ${\mathbf{k}}_{\mathrm{rel}}$ and center-of-mass (c.m.) momentum ${\mathbf{K}}_{\mathrm{c}.\mathrm{m}.}$; at high values of the relative momentum and, at the same time, low values of the c.m. momentum, $n_A^{N_1{N}_2}({\mathbf{k}}_{\mathrm{rel}},{\mathbf{K}}_{\mathrm{c}.\mathrm{m}.})$ provides information on the short-range structure of nuclei.

Purpose: Our purpose is to calculate the momentum distributions of proton-neutron and proton-proton pairs in ${}^3\mathrm{He}, {}^4\mathrm{He}, {}^{12}\mathrm{C}, {}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$, in correspondence to various values of ${\mathbf{k}}_{\mathrm{rel}}$ and ${\mathbf{K}}_{\mathrm{c}.\mathrm{m}.}$.

Methods: The momentum distributions for $A>4$ nuclei are calculated as a function of the relative, $k_{\text{rel}}$, and center-of-mass, $K_{\text{c.m.}}$, momenta and relative angle $\mathrm{\Theta }$, within a linked cluster many-body expansion approach, based upon realistic local two-nucleon interaction of the Argonne family and variational wave functions featuring central, tensor, and spin-isospin correlations.

Results: Independently of the mass number $A$, at values of the relative momentum $k_{\text{rel}}\gtrsim 1.5–2 {\mathrm{fm}}^{-1}$ the momentum distributions exhibit the property of factorization, $n_A^{N_1{N}_2}({\mathbf{k}}_{\mathrm{rel}},{\mathbf{K}}_{\mathrm{c}.\mathrm{m}.})\simeq n_{\mathrm{rel}}^{N_1{N}_2}\left(k_{\mathrm{rel}}\right)n_{\mathrm{c}.\mathrm{m}.}^{N_1{N}_2}\left(K_{\mathrm{c}.\mathrm{m}.}\right)$; in particular, for $pn$ back-to-back pairs one has $n_A^{pn}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.}=0)\simeq C_A^{pn}{n}_D\left(k_{\mathrm{rel}}\right)n_{\mathrm{c}.\mathrm{m}.}^{pn}(K_{\mathrm{c}.\mathrm{m}.}=0)$, where $n_D$ is the deuteron momentum distribution, $n_{\mathrm{c}.\mathrm{m}.}^{pn}(K_{\mathrm{c}.\mathrm{m}.}=0)$ the c.m. motion momentum distribution of the pair, and $C_A^{pn}$ the $pn$ nuclear contact measuring the number of back-to-back $pn$ pairs with deuteron-like momenta (${\mathbf{k}}_p\simeq -{\mathbf{k}}_n,{\mathbf{K}}_{\mathrm{c}.\mathrm{m}.}=0$).

Conclusions: The values of the $pn$ nuclear contact are extracted from the general properties of the two-nucleon momentum distributions corresponding to $K_{\mathrm{c}.\mathrm{m}.}=0$. The $K_{\mathrm{c}.\mathrm{m}.}$-integrated $pn$ momentum distributions exhibit the property $n_A^{pn}\left(k_{\mathrm{rel}}\right)\simeq C_A^{pn}{n}_D\left(k_{\mathrm{rel}}\right)$ but only at very high values of $k_{\mathrm{rel}}, \gtrsim 3.5–4 {\mathrm{fm}}^{-1}$. The theoretical ratio of the $pp/pn$ momentum distributions of ${}^4\mathrm{He}$ and ${}^{12}\mathrm{C}$ and the calculated c.m. motion momentum distributions are in agreement with recent experimental data.

Figure 1

(a) Two-nucleon momentum distributions of $pn$ pairs in ${}^3\mathrm{He}$ vs the relative momentum $k_{\text{rel}}$ for fixed values of the c.m. momentum $K_{\text{c.m.}}$ (in ${\mathrm{fm}}^{-1}$) and two values of the angle $\mathrm{\Theta }$ between ${\mathbf{k}}_{\mathrm{rel}}$ and ${\mathbf{K}}_{\text{c.m.}}$, namely, $\mathrm{\Theta }={90}^{\circ }$ (broken curves) and $\mathrm{\Theta }=0^{\circ }$ (symbols). In this figure, and only in it, solid curves represent Eq. (10) with $C_3^{pn}=2.0$. ${}^3\mathrm{He}$ wave function from Refs. [32, 33] and AV18 interaction [35]. (b) Same as (a), but for $pp$ pairs. (c) The $pn$ and $pp$ distributions corresponding to $K_{\mathrm{c}.\mathrm{m}.}=0$ in (a) and (b) and their sum. (d) Relative two-body momentum distributions $n_A^{N_1{N}_2}\left(k_{\mathrm{rel}}\right)=\int n_A^{N_1{N}_2}({\mathbf{k}}_{\mathrm{rel}},{\mathbf{K}}_{\mathrm{c}.\mathrm{m}.})d{\mathbf{K}}_{\mathrm{c}.\mathrm{m}.}$. In (c) open and filled circles represent the results from Argonne [11]. In this and the following figures, unless differently stated, $pn$ and $pp$ distributions are normalized to $ZN$ and $Z(Z-1)/2$, respectively.

Figure 2

Same as Fig. 1, but for ${}^4\mathrm{He}$ with $C_4^{pn}=4.0$ in (a). ${}^4\mathrm{He}$ wave function from Ref. [34] and AV8′ interaction [36]. In (c) and (d) open and filled circles denote the results from Argonne [11].

Figure 3

Same as Fig. 1, but for ${}^{12}\mathrm{C}$ with $C_{12}^{pn}=20.0$ in (a). ${}^{12}\mathrm{C}$ wave function from Ref. [15] and AV8′ interaction [36]. Note that here, as well as in Figs. 4 and 5, symbols in (a) correspond to $\mathrm{\Theta }=0$. In (c) open and filled circles represent the results from Argonne [11].

Figure 4

Same as Fig. 1, but for ${}^{16}\mathrm{O}$ with $C_{16}^{pn}=24.0$ in (a). ${}^{16}\mathrm{O}$ wave function from Ref. [15] and AV8′ interaction [36].

Figure 5

Same as Fig. 1, but for ${}^{40}\mathrm{Ca}$ with $C_{40}^{np}=60.0$ in (a). ${}^{40}\mathrm{Ca}$ wave function from Ref. [15] and AV8′ interaction [36].

Figure 6

(a) The center-of-mass (c.m.) momentum distribution $n_{\text{A}}^{pn}\left(K_{\text{c.m.}}\right)=\int n_A^{pn}({\mathbf{k}}_{\text{rel}},{\mathbf{K}}_{\text{c.m.}})d{}^3{\mathbf{k}}_{\text{rel}}$ [Eq. (8)] in ${}^3\mathrm{He}, {}^4\mathrm{He}, {}^{12}\mathrm{C}$, and ${}^{40}\mathrm{Ca}$, normalized to 1, obtained in the present paper (this work), in Ref. [25] (CS), and in Ref. [11] (Argonne). Note that a Gaussian distribution related to the average value of the shell-model kinetic energy [25] agrees very well with the many-body realistic distribution up to $K_{\mathrm{c}.\mathrm{m}.}\simeq 1{\mathrm{fm}}^{-1}$ except in the case of ${}^3\mathrm{H}$, for which a shell-model description has no meaning. (b) The c.m. momentum distributions of ${}^3\mathrm{He}, {}^{12}\mathrm{C}$, and ${}^{40}\mathrm{Ca}$ on a linear scale.

Figure 7

Comparison of the one-nucleon momentum distributions of ${}^4\mathrm{He}$ and $\mathrm{C}$ calculated in Ref. [17] (solid line) and in Ref. [11] (open circles), respectively. Normalization to the number of protons $Z$.

Figure 8

Determining the constant $C_A^{pn}$ by a plot of Eq. (12) vs $k_{\mathrm{rel}}$ with a fixed value of $K_{\mathrm{c}.\mathrm{m}.}=0$. The constant value of Eq. (12) determines the value of $C_A^{pn}$. For ${}^3\mathrm{He}, {}^4\mathrm{He}, {}^6\mathrm{Li}$, and ${}^8\mathrm{Be}$ the results obtained with the Argonne momentum distributions [11] are also shown.

Figure 9

Ratio [Eq. (16)] between the factorized distributions [Eq. (10)] and the exact ones ($\mathrm{\Theta }=0^{\circ }$) for ${}^4\mathrm{He}, {}^{12}\mathrm{C}, {}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$ in correspondence to $K_{\mathrm{c}.\mathrm{m}.}=0$, 0.5, and 1 ${\mathrm{fm}}^{-1}$. For ${}^4\mathrm{He}$ the results obtained with the Argonne momentum distributions [11] are shown by the solid line.

Figure 10

Ratio [Eq. (16)] between the factorized distributions, Eq. (10), and the exact ones for ${}^6\mathrm{Li}$ and ${}^8\mathrm{Be}$, corresponding to $K_{\mathrm{c}.\mathrm{m}.}=0$, obtained with the Argonne VMC wave functions [11].

Figure 11

(a), (b) Comparison of the deuteron momentum distributions with the two-nucleon momentum distributions of various nuclei obtained in the present paper. (a) The validity of the relation $n_A^{pn}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.})/C_A^{pn}{n}_{\mathrm{c}.\mathrm{m}.}^{pn}(K_{\mathrm{c}.\mathrm{m}.}=0)\simeq n_D\left(k_{\mathrm{rel}}\right)$, when $K_{\mathrm{c}.\mathrm{m}.}=0$ and $k_{\mathrm{rel}}\ge \sim 2{\mathrm{fm}}^{-1}$, is demonstrated. (b) Demonstration that when the $K_{\mathrm{c}.\mathrm{m}.}$-integrated two-nucleon momentum distributions are considered, the relation $n_A^{pn}\left(k_{\mathrm{rel}}\right)/C_A^{pn}\simeq n_D\left(k_{\mathrm{rel}}\right)$ is also valid, but only at $k_{\mathrm{rel}}\ge 3.5–4 {\mathrm{fm}}^{-1}$. (c), (d) The quantity $n_A^{pn}\left(k_{\mathrm{rel}}\right)/C_A^{pn}$ for ${}^4\mathrm{He}$ (c) and ${}^{12}\mathrm{C}$ corresponding to the momentum distributions obtained in the present paper and in Ref. [11]. In (c) and (d) the values of $C_A^{pn}$ are the ones listed in Table 1. These results unambiguously prove that both $n_A^{pn}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.}=0)$ and $n_A^{pn}\left(k_{\mathrm{rel}}\right)$ do factorize to the deuteron momentum distribution, but starting at appreciably different values of $k_{\mathrm{rel}}$ in the two cases. These results also show that $n_A^{pn}\left(k_{\mathrm{rel}}\right)$ at $k_{\mathrm{rel}}\ge 3.5–4 {\mathrm{fm}}^{-1}$ is mainly governed by back-to-back $pn$ pairs.

Figure 12

(a) The exact proton momentum distribution $n_A^p\left(k\right)$ ($k_1\equiv k$) compared with the convolution model, Eq. (21) (Conv), calculated with two different expressions for $n_{\mathrm{rel}}^{pn}$. (b) The exact $n_A^p\left(k\right)$ compared with (i) the asymptotic approximation of the convolution model [Eqs. (21) and (22)] ($K_{\mathrm{c}.\mathrm{m}.}=0$); (ii) Eq. (21) with $n_{\mathrm{rel}}^{N_1{N}_2}(k_1=k_{\mathrm{rel}})$ replaced by the $K_{\mathrm{c}.\mathrm{m}.}$-integrated relative momentum distributions $n_A^{N_1{N}_2}\left(k_{\mathrm{rel}}\right)=\int n_A^{N_1{N}_2}({\mathbf{k}}_{\mathrm{rel}},{\mathbf{K}}_{\mathrm{c}.\mathrm{m}.})d{\mathbf{K}}_{\mathrm{c}.\mathrm{m}.}$; and (iii) the convolution model [Eq. (24)] as in (a).

Figure 13

Ratio of the full two-nucleon momentum distribution $n_{pn}+2n_{pp}$ to the one-nucleon momentum distribution $n_A^p$ for ${}^4\mathrm{He}$ (a) and ${}^{12}\mathrm{C}$ (b) calculated using in the numerator the relative two-nucleon distributions $n_A^{pn}(k_{\mathrm{rel}}=k_1,K_{\mathrm{c}.\mathrm{m}.}=0)/n_{\mathrm{c}.\mathrm{m}.}(K_{\mathrm{c}.\mathrm{m}.}=0)$ [Eq. (24); open circles] and the ${\mathbf{K}}_{\mathrm{c}.\mathrm{m}.}$-integrated two-nucleon momentum distributions $n_A^{pn}\left(k_{\mathrm{rel}}\right)$ [Eq. (25); filled squares]; in both cases the denominator is the one-nucleon momentum distribution. The solid line denotes the results obtained with the Argonne VMC wave function [11].

Figure 14

(a) The ratio $N_{pn}^{\mathrm{SRC},\mathrm{BB}}/N_{pp}^{\mathrm{SRC},\mathrm{BB}}$ using the wave functions of the present work (cf. Table 2) and the VMC results from Ref. [11]. (b) The same as (a), for the $K_{\mathrm{cm}}$-integrated momentum distributions (cf. Table 3).

Figure 15

(a) The experimental percentage of the $pN$ BB pair fraction $pp/pn$ vs the missing momentum $p_m$, extracted from the processes ${}^4\mathrm{He}(e,e^{\prime }pn)X$ [30] and ${}^{12}\mathrm{C}(e,e^{\prime }pp)X$ [28, 29] compared with the quantity $R_{pp/pn}(k_{\mathrm{rel}},0)=n_{pp}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.}=0)/n_{pn}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.}=0)$ calculated in the present paper (solid line). Open squares show the results obtained with the Argonne momentum distributions [11]. (b) The same ratio as in (a), calculated within two approaches: (i) solid line, $n_{A=4}^{pp}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.}=0)/n_{A=4}^{pn}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.}=0)$; (ii) dashed line, $\frac{{\int }_0^{\infty }{n}_{pp}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.})K_{\mathrm{c}.\mathrm{m}.}^{2}d{K}_{\mathrm{c}.\mathrm{m}.}}{{\int }_0^{\infty }{n}_{pn}(k_{\mathrm{rel}},K_{\mathrm{c}.\mathrm{m}.})K_{\mathrm{c}.\mathrm{m}.}^{2}d{K}_{\mathrm{c}.\mathrm{m}.}}$.

Figure 16

Experimental percentage of SRC fractions in ${}^4\mathrm{He}$ (a) and ${}^{12}\mathrm{C}$ (b) compared with theoretical ratios of momentum distributions within the assumption ${\mathbf{p}}_m\simeq {\mathbf{k}}_{\mathrm{rel}}$ and $K_{\mathrm{c}.\mathrm{m}.}=0$. Momentum distributions from the present work and from the Argonne VMC calculation [11]. All experimental data are from Jlab [27, 28, 29, 30, 31], except the one represented by the magenta point for ${}^{12}\mathrm{C}$, which was obtained from the BNL [26]. In (b) the three theoretical curves were obtained in the present work and correspond to $pp/p$ (solid line), $pn/p$ (dashed line), and $pp/pn$ (dotted line), respectively.

Figure 17

The c.m. momentum distribution of a $pn$ pair in ${}^4\mathrm{He}$ (a) and a $pp$ pair in ${}^{12}\mathrm{C}$ (b) obtained in Refs. [30] and [28] from the processes ${}^4\mathrm{He}(e,e^{\prime }pn)X$ and ${}^{12}\mathrm{C}(e,e^{\prime }pp)X$. $\gamma$ is the angle between ${\mathbf{p}}_m$ and ${\mathbf{p}}_{\mathrm{rec}}$, which in the PWIA is the angle between ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$. The values of ${\mathbf{K}}_{\mathrm{c}.\mathrm{m}.}$ have been obtained assuming ${\mathbf{k}}_2=-{\mathbf{k}}_1$. The theoretical curves correspond to the momentum distributions in Ref. [11] (Argonne) and Ref. [25] (CS). Experimental data are given in arbitrary units and theoretical calculations were normalized at the lowest available experimental point. Note that the discrepancy or the agreement between the experimental data and the theoretical calculations might not be real ones, since the latter, unlike the former, do not take into account the finite acceptance and resolution of the detectors and the continuous background. Indeed, when these are taken into account by a proper simulation [28, 30, 41], the data can be explained by the Gaussian distribution $n_{\mathrm{c}.\mathrm{m}.}\left(K_{\mathrm{c}.\mathrm{m}.}\right)={(\alpha /\pi )}^{1.5}exp(-\alpha K_{\mathrm{c}.\mathrm{m}.}^2)$ in agreement with the three curves in (a) and (b).